Two-parameter regularization method for an axisymmetric inverse heat problem
- Ngo Van Hoa^{1, 2}Email author and
- Tra Quoc Khanh^{3}
Received: 8 September 2016
Accepted: 16 January 2017
Published: 9 February 2017
Abstract
In this paper we consider the inverse time problem for the axisymmetric heat equation which is a severely ill-posed problem. Using the modified quasi-boundary value (MQBV) method with two regularization parameters, one related to the error in measurement process and the other related to the regularity of solution, we regularize this problem and obtain the Hölder-type estimation error for the whole time interval. Numerical results are presented to illustrate the accuracy and efficiency of the method.
Keywords
1 Introduction
Partial differential equations (PDEs) associated with various types of boundary conditions are a powerful and useful tool to model natural phenomena. For time-dependent phenomena, they are usually joined by a time condition (initial time condition or final time condition), which can be considered as the data. The time-inverse problem means that, from the final data, the main goal is a reconstruction of the whole structure in previous time. These problems were widely investigated in Tikhonov and Arsenin [1] and Glasko [2]. A classical example can be recalled here: the backward heat conduction problem (BHCP). The BHCP is the time-inverse boundary value problem, i.e., given the information at a specific point of time, say \(t=T\), the goal is to recover the corresponding structure at an earlier time \(t< T\).
The BHCP is strictly difficult to solve since, in general, the solution does not always exist. Furthermore, even if the solution does exist, it would not be continuously dependent on the data. As a result, there are a number of difficulties in doing numerical calculations. BHCP is a very famous problem and has been considered by many authors by different methods [3–19]. For the BHCP with a constant coefficient, there is much nice literature that can be listed. Trong and Tuan in [12] used the method of integral equation to regularize backward heat conduction problem and get some error estimates. In [16], Hao et al. gave a very nice approximation for this problem by using a non-local boundary value problem method. Later on, Hao and Duc [17] used the Tikhonov regularization method to give an approximation for this problem in Banach space. Tautenhahn in [10] established an optimal error estimate for a backward heat equation with constant coefficient. Fu et al. [7] applied a wavelet dual least squares method to investigate a BHCP with constant coefficient.
2 Statement of the problem
Let us first make it clear what a solution of the problem (1.1) is. By a solution of (1.1), we imply a function \(u(r,t)\) satisfying (1.1) in the classical sense and for every fixed \(t \in[0, T]\) and this function \(u(r,t) \in L^{2}(0,r_{0};r) \). In this class of functions, if the solution of problem (1.1) exists, then it must be unique (see [21]).
Theorem 1
Cheng and Fu [18]
Proof
Regarding (2.11), the exponential growth causes an instability in the solution, i.e., the problem is ill-posed and a regularization method is extremely important.
3 Regularization and error estimates
Lemma 1
Proof
The following theorem shows the well-posedness of the regularized problem (3.1).
Theorem 2
Proof
The proof is divided into two steps. In Step 1, we prove the existence and the uniqueness of solution of the regularized problem (3.1). In Step 2, the stability of the solution is given.
Until now, we already stated that our regularized problem is a well-posed problem in the sense of Hadamard. In the following, we will establish an error estimate between the exact solution and the regularized solution.
Theorem 3
The error estimate in the case of exact data
Proof
Theorem 4
Error estimates in the case of non-exact data
Proof
Remark 1
4 A numerical illustration
Fix \(T=1\), \(r_{0}=2\), \(P_{0} = 1\text{,}000\), \(A^{2} = 100\). There are two situations to consider.
The error and relative error of method in this paper with \(\pmb{\tau=0.3}\) and various values of ε
ε | \(\boldsymbol{\Vert {{u^{{\varepsilon ,0.3 }}}(\cdot,0) - {u^{ex}}(\cdot,0)} \Vert }\) | RE( ε ,0) |
---|---|---|
ε = 10^{−3} | 8.48702923827765 | 0.115626406608408 |
ε = 10^{−4} | 5.84620043944197 | 0.079648028791562 |
ε = 10^{−5} | 3.59734075211554 | 0.049009797519858 |
ε = 10^{−6} | 0.723559578438893 | 0.009857700695156 |
ε = 10^{−7} | 0.0550576642083171 | 0.000173396997915 |
Remark 2
Through Figure 1, Figure 2, Figure 3, and Table 1, it is clear that as the measuring error ε becomes small, the regularized solution gets ever more close to the exact one. It is also noted that the value of \(a_{p}\) ranges from −307.896 to 290.9509 in this situation.
Remark 3
Figure 4 and Figure 5 agree with the theoretical result in Section 3: the regularized solution with higher value of τ is more close to the exact one. The parameter τ is very useful in the case that we want to get a more accurate approximation while the measuring process cannot be better or the cost of better measuring is very expensive. In this case, with the appearance of τ, the error can be improved without any more cost on measuring (as we can see in Figure 5). It is also noted that the value of \(a_{p}\) ranges from −309.433 to 384.8187 in this situation.
Declarations
Acknowledgements
The authors would like to thank Professor Dang Duc Trong and Associate Professor Nguyen Huy Tuan for the great support during their undergraduate period.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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