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

Boundary Value Problems20092009:260941

DOI: 10.1155/2009/260941

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 [113]. These methods include Tikhonov method [1, 2], mollification method [3, 4], optimal filtering method [5], lines method [6], wavelet and wavelet-Galerkin method [711], 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 [1419].

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.

In this paper, we will treat with a special IHCP whose physical model consists of an infinitely long cylinder of radius http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq1_HTML.gif . It is considered axisymmetric and a thermocouple (measurement equipment of temperature) is installed inside the cylinder (at the radius http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq3_HTML.gif ). The correspondingly mathematical model of our problem can be described by the following axisymmetric heat conduction problem:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ1_HTML.gif
(1.1)

where the functions http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq5_HTML.gif belong to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq6_HTML.gif for every fixed http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq8_HTML.gif is the radial coordinate, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq9_HTML.gif denotes the temperature history at one fixed radius http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq10_HTML.gif of cylinder. We want to recover http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq11_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq12_HTML.gif . This problem is ill-posed problem; a small perturbation in the data may cause dramatically large errors in the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq13_HTML.gif (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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq14_HTML.gif . 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)

As we consider problem (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq15_HTML.gif with respect to variable http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq16_HTML.gif , we extend http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq17_HTML.gif , and other functions of variable http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq18_HTML.gif appearing in the paper to be zero for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq19_HTML.gif . Throughout the paper, we assume that for the exact http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq20_HTML.gif the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq21_HTML.gif exists and satisfies an apriori bound
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ2_HTML.gif
(2.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq22_HTML.gif is defined by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ3_HTML.gif
(2.2)
Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq23_HTML.gif is measured by the thermocouple, there will be measurement errors, and we would actually have as data some function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq24_HTML.gif , for which
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ4_HTML.gif
(2.3)
where the constant http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq25_HTML.gif represents a bound on the measurement error, and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq26_HTML.gif denotes the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq27_HTML.gif norm and
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ5_HTML.gif
(2.4)
is the Fourier transform of function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq28_HTML.gif . The problem (1.1) can be formulated, in frequency space, as follows:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ6_HTML.gif
(2.5)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ7_HTML.gif
(2.6)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ8_HTML.gif
(2.7)

Then we have the following lemma.

Lemma 2.1.

Problem (2.5)–(2.7) has the solution given by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ9_HTML.gif
(2.8)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq29_HTML.gif denotes modified spherical Bessel function which given by [21]
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ10_HTML.gif
(2.9)

Proof.

Due to [21], we can solve (2.5), in the frequency domain, to obtain
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ11_HTML.gif
(2.10)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq30_HTML.gif denotes also modified spherical Bessel function which is given by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ12_HTML.gif
(2.11)
Combining http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq31_HTML.gif with condition(2.7), we obtain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq32_HTML.gif , that is,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ13_HTML.gif
(2.12)
According to [21], there holds
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ14_HTML.gif
(2.13)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq33_HTML.gif , both http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq35_HTML.gif denote the Kelvin functions. Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq36_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ15_HTML.gif
(2.14)
Therefore, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq37_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ16_HTML.gif
(2.15)
Solving the systems (2.6) and (2.12) using (2.15) we get
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ17_HTML.gif
(2.16)

Substitution of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq38_HTML.gif in (2.16) into (2.12), we obtain (2.8).

Applying an inverse Fourier transform to (2.8), problem (1.1) has the solution
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ18_HTML.gif
(2.17)

In order to obtain ill-posedness of problem (1.1) for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq39_HTML.gif , we need the following lemma.

Lemma 2.2.

If function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq40_HTML.gif satisfies (2.15), then there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq41_HTML.gif such that, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq42_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ19_HTML.gif
(2.18)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ20_HTML.gif
(2.19)

Proof.

First, due to [21] and (2.15), we have, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq43_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq44_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ21_HTML.gif
(2.20)
then there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq45_HTML.gif such that, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq46_HTML.gif large enough, say http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq47_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ22_HTML.gif
(2.21)
From these we know that there exist positive constants http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq48_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq49_HTML.gif such that, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq50_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq51_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ23_HTML.gif
(2.22)
Then, since function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq52_HTML.gif is continuous in the closed region http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq53_HTML.gif . Threrfore, there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq54_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq55_HTML.gif such that, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq56_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq57_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ24_HTML.gif
(2.23)

Finally, combining inequalities (2.22) with (2.23), we can see that there exist others constants http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq59_HTML.gif such that, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq60_HTML.gif , inequalities (2.18) are valid. Similarly, we obtain inequalities (2.19).

In order to formulate problem (1.1) for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq61_HTML.gif in terms of an operator equation in the space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq62_HTML.gif , we define an operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq63_HTML.gif , that is,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ25_HTML.gif
(2.24)
From (2.8), we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ26_HTML.gif
(2.25)
Denote http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq64_HTML.gif , and we can see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq65_HTML.gif is a multiplication operator:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ27_HTML.gif
(2.26)

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

Lemma 2.3.

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq66_HTML.gif be the adjoint to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq67_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq68_HTML.gif corresponds to the following problem where the left-hand side http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq69_HTML.gif of problem (1.1) is replaced by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq70_HTML.gif , says
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ28_HTML.gif
(2.27)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ29_HTML.gif
(2.28)

Proof.

Via the the following relations, combining with (2.26),
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ30_HTML.gif
(2.29)
we can get the adjoint operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq71_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq72_HTML.gif in frequency domain
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ31_HTML.gif
(2.30)
On the other hand, the problem (2.27) can be formulated, in frequency space, as follows:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ32_HTML.gif
(2.31)
Taking the conjugate operator for problem (2.5)–(2.7), we realize that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq73_HTML.gif . Therefore, by Lemma 2.1, we conclude that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ33_HTML.gif
(2.32)
that is,
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ34_HTML.gif
(2.33)

Hence the conclusion of Lemma 2.3 is proved.

The Parseval formula for the Fourier transform together with inequality (2.18), there holds
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ35_HTML.gif
(2.34)

This implies that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq74_HTML.gif , which is Fourier transform of exact data http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq75_HTML.gif , must decay rapidly at high frequencies since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq76_HTML.gif . But such a decay is not likely to occur in the Fourier transform of the measured noisy data http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq77_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq78_HTML.gif . So, small perturbation of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq79_HTML.gif in high frequency components can blow up and completely destroy the solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq80_HTML.gif given by (2.17) for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq81_HTML.gif .

3. Wavelet Dual Least Squares Method

3.1. Dual Least Squares Method

A general projection method for the operator equation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq82_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq83_HTML.gif is generated by two subspace families http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq84_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq85_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq86_HTML.gif and the approximate solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq87_HTML.gif is defined to be the solution of the following problem:
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ36_HTML.gif
(3.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq88_HTML.gif denotes the inner product in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq89_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq90_HTML.gif and subspaces http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq91_HTML.gif are chosen in such a way that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ37_HTML.gif
(3.2)
Then we have a special case of projection method known as the dual least squares method. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq92_HTML.gif is an orthogonal basis of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq93_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq94_HTML.gif is the solution of the equation
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ38_HTML.gif
(3.3)
then the approximate solution is explicitly given by the expression
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ39_HTML.gif
(3.4)

3.2. Shannon Wavelets

In [22], the Shannon scaling function is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq95_HTML.gif and its Fourier transform is
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ40_HTML.gif
(3.5)
The corresponding wavelet function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq96_HTML.gif is given by its Fourier transform
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ41_HTML.gif
(3.6)
Let us list some notation: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq97_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq98_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq99_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq101_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq102_HTML.gif , the index set
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ42_HTML.gif
(3.7)
Because http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq103_HTML.gif , hence we can define the subspaces http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq104_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ43_HTML.gif
(3.8)
Define an orthogonal projection http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq105_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ44_HTML.gif
(3.9)
then from (3.4) we easily conclude http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq106_HTML.gif . From the point of view of an application to the problem (1.1), the important property of Shannon wavelets is the compactness of their support in the frequency space. Indeed, since
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ45_HTML.gif
(3.10)
it follows that for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq107_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ46_HTML.gif
(3.11)

From (3.9), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq108_HTML.gif can be seen as a low-pass filter. The frequencies with greater than http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq109_HTML.gif are filtered away.

Theorem 3.1.

If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq110_HTML.gif is the solution of problem (1.1) satisfying the condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq111_HTML.gif , then for any fixed http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq112_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ47_HTML.gif
(3.12)

Proof.

From (3.9), we have
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ48_HTML.gif
(3.13)
Due to Parseval relation and (2.8), (2.19), and (2.1), there holds
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ49_HTML.gif
(3.14)

Hence the conclusion of Theorem3.1 is proved.

4. Error Estimates via Dual Least Squares Method Approximation

Before giving error estimates, we present firstly subspaces http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq113_HTML.gif . According to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq114_HTML.gif , the subspaces http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq115_HTML.gif are spanned by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq116_HTML.gif , where
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ50_HTML.gif
(4.1)
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq117_HTML.gif can be determined by solving the following parabolic equation (see Lemma 2.3):
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ51_HTML.gif
(4.2)
Since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq118_HTML.gif is compact, the solution exists for any http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq119_HTML.gif . Similarly the solution of the adjoint equation is unique. Therefore for a given http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq120_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq121_HTML.gif can be uniquely determined according to (4.2), furthermore
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ52_HTML.gif
(4.3)
The approximate solution for noisy data http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq122_HTML.gif is explicitly given by
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ53_HTML.gif
(4.4)

Now we will devote to estimating the error http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq123_HTML.gif .

Theorem 4.1.

If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq124_HTML.gif is noisy data satisfying the condition http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq125_HTML.gif , then for any fixed http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq126_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ54_HTML.gif
(4.5)

Proof.

From (4.3), we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq127_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq128_HTML.gif given by (4.4), http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq129_HTML.gif given by (3.4) and (2.18), for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq130_HTML.gif , there holds
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ55_HTML.gif
(4.6)

Hence the conclusion of Theorem 4.1 is proved.

The following is the main result of this paper.

Theorem 4.2.

Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq131_HTML.gif be the exact solution of (1.1) and let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq132_HTML.gif be given by (4.4). If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq133_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq134_HTML.gif is such that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ56_HTML.gif
(4.7)
then for any fixed http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq135_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ57_HTML.gif
(4.8)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq136_HTML.gif .

Proof.

Combining Theorem 4.1 with Theorem 3.1, and noting the choice rule (4.7) of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq137_HTML.gif , we can obtain
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ58_HTML.gif
(4.9)
Note that
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ59_HTML.gif
(4.10)
thus, there holds, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq138_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ60_HTML.gif
(4.11)

Hence the conclusion of Theorem 4.2 is proved.

Remark 4.3.
  1. (i)
    When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq139_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq140_HTML.gif , estimate (4.8) is a Hölder stability estimate given by
    http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ61_HTML.gif
    (4.12)
     
  1. (ii)

    When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq141_HTML.gif , estimate (4.8) is a logarithmical Hölder stability estimate.

     
  2. (iii)
    When http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq142_HTML.gif , estimate (4.3) becomes
    http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ62_HTML.gif
    (4.13)
     

This is a logarithmical stability estimate.

Remark 4.4.

In general, the a-priori bound http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq143_HTML.gif is unknown in practice, in this case, with
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ63_HTML.gif
(4.14)
then
http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_Equ64_HTML.gif
(4.15)

where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F260941/MediaObjects/13661_2008_Article_835_IEq144_HTML.gif .

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

(1)
College of Science, Henan University of Technology
(2)
School of Mathematics and Statistics, Lanzhou University

References

  1. Carasso A: Determining surface temperatures from interior observations. SIAM Journal on Applied Mathematics 1982, 42(3):558–574. 10.1137/0142040MATHMathSciNetView Article
  2. 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
  3. 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
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. 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
  19. 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
  20. 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
  21. Abramowitz M, Stegun IA (Eds): Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover, New York, NY, USA; 1972.MATH
  22. 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

© W. Cheng and C.-L. Fu. 2009

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.