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.
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 , lines method , wavelet and wavelet-Galerkin method [7–11], modified Tikhonov method  and "optimal approximations" , 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.  applied the hybrid numerical algorithm of Laplace transform technique to the IHCP in a rectangular plate. Busby and Trujillo  used the dynamic programming method to investigate the IHCP in a slab. Alifanov and Kerov  and Louahlia-Gualous et al.  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 . It is considered axisymmetric and a thermocouple (measurement equipment of temperature) is installed inside the cylinder (at the radius , ). The correspondingly mathematical model of our problem can be described by the following axisymmetric heat conduction problem:
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  and Regińska  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 with respect to variable , we extend , and other functions of variable appearing in the paper to be zero for . Throughout the paper, we assume that for the exact the solution exists and satisfies an apriori bound
where is defined by
Since is measured by the thermocouple, there will be measurement errors, and we would actually have as data some function , for which
where the constant represents a bound on the measurement error, and denotes the norm and
is the Fourier transform of function . The problem (1.1) can be formulated, in frequency space, as follows:
Then we have the following lemma.
Problem (2.5)–(2.7) has the solution given by
where denotes modified spherical Bessel function which given by 
Due to , we can solve (2.5), in the frequency domain, to obtain
where denotes also modified spherical Bessel function which is given by
Combining with condition(2.7), we obtain , that is,
According to , there holds
where , both and denote the Kelvin functions. Since , we have
Therefore, for ,
Solving the systems (2.6) and (2.12) using (2.15) we get
Substitution of in (2.16) into (2.12), we obtain (2.8).
Applying an inverse Fourier transform to (2.8), problem (1.1) has the solution
In order to obtain ill-posedness of problem (1.1) for , we need the following lemma.
If function satisfies (2.15), then there exist positive constants such that, for
First, due to  and (2.15), we have, for and ,
then there exist positive constants such that, for large enough, say
From these we know that there exist positive constants and such that, for and ,
Then, since function is continuous in the closed region . Threrfore, there exist constants and such that, for and ,
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).
In order to formulate problem (1.1) for in terms of an operator equation in the space , we define an operator , that is,
From (2.8), we have
Denote , and we can see that is a multiplication operator:
From (2.26), we can prove the following lemma.
Let be the adjoint to , then corresponds to the following problem where the left-hand side of problem (1.1) is replaced by , says
Via the the following relations, combining with (2.26),
we can get the adjoint operator of in frequency domain
On the other hand, the problem (2.27) can be formulated, in frequency space, as follows:
Taking the conjugate operator for problem (2.5)–(2.7), we realize that . Therefore, by Lemma 2.1, we conclude that
Hence the conclusion of Lemma 2.3 is proved.
The Parseval formula for the Fourier transform together with inequality (2.18), there holds
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
A general projection method for the operator equation , is generated by two subspace families and of and the approximate solution is defined to be the solution of the following problem:
where denotes the inner product in . If and subspaces are chosen in such a way that
Then we have a special case of projection method known as the dual least squares method. If is an orthogonal basis of and is the solution of the equation
then the approximate solution is explicitly given by the expression
3.2. Shannon Wavelets
In , the Shannon scaling function is and its Fourier transform is
The corresponding wavelet function is given by its Fourier transform
Let us list some notation: , , , and for , the index set
Because , hence we can define the subspaces
Define an orthogonal projection :
then from (3.4) we easily conclude . 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
it follows that for any
From (3.9), can be seen as a low-pass filter. The frequencies with greater than are filtered away.
If is the solution of problem (1.1) satisfying the condition , then for any fixed
From (3.9), we have
Due to Parseval relation and (2.8), (2.19), and (2.1), there holds
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 . According to , the subspaces are spanned by , where
can be determined by solving the following parabolic equation (see Lemma 2.3):
Since is compact, the solution exists for any . Similarly the solution of the adjoint equation is unique. Therefore for a given , can be uniquely determined according to (4.2), furthermore
The approximate solution for noisy data is explicitly given by
Now we will devote to estimating the error .
If is noisy data satisfying the condition , then for any fixed
From (4.3), we have . Note that given by (4.4), given by (3.4) and (2.18), for , there holds
Hence the conclusion of Theorem 4.1 is proved.
The following is the main result of this paper.
Let be the exact solution of (1.1) and let be given by (4.4). If and is such that
then for any fixed
Combining Theorem 4.1 with Theorem 3.1, and noting the choice rule (4.7) of , we can obtain
thus, there holds, for
Hence the conclusion of Theorem 4.2 is proved.
(i) When and , estimate (4.8) is a Hölder stability estimate given by
(ii) When , estimate (4.8) is a logarithmical Hölder stability estimate.
(iii) When , estimate (4.3) becomes
This is a logarithmical stability estimate.
In general, the a-priori bound is unknown in practice, in this case, with
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).
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