Cauchy problem for the Laplace equation in a radially symmetric hollow cylinder
- Yun-Jie Ma^{1}Email author and
- Chu-Li Fu^{2}
Received: 20 April 2016
Accepted: 25 October 2016
Published: 8 November 2016
Abstract
In this paper, an axisymmetric Cauchy problem for the Laplace equation in an unbounded hollow cylinder is considered. The Cauchy data are given on the inside surface of the cylinder, and the solution on the whole domain is sought. We propose a Fourier method with a priori and a posteriori parameter choice rules to solve this ill-posed problem. It is shown that the approximate solutions are stably convergent to the exact ones with explicit error estimates. A further comparison in the numerical aspects demonstrates the effectiveness and accuracy of the presented methods.
Keywords
MSC
1 Introduction
In this paper, we consider the problem of an extension of the field potential specified on the inside surface of a hollow cylinder into space, and it is reduced to the axisymmetric Cauchy problem for the Laplace equation. This problem is involved in practical calculations of various electron optic systems. The hollow cylinder case is interesting since the hole leaves spaces for the measurement devices or devices that generate electric or magnetic fields. For example, electric fields with rotational symmetry are usually generated by electrodes in the shape of cylinders, cups and diaphragms. In recent years, Lu et al. [11] applied an analytical approach to study the transient heat conduction in a composite hollow cylinder. Cheng et al. [12] studied the inverse heat conduction problem in a hollow spherically symmetric domain. Marin and Marinescu [13, 14] investigated the existence, uniqueness and the asymptotic partition of total energy for the solutions of the initial boundary value problem within the context of the thermoelasticity of initially stressed bodies, and further considered micropolar thermoelastic body occupying a prismatic cylinder [15]. Şeremet and Şeremet [16] presented new steady-state Green’s functions for displacements and thermal stresses for plane problem within a rectangular region, and the proposed technique could be extended to many 3D problems. More detailed descriptions of the model of hollow cylinders can be found in [17].
This paper is organized as follows. In Section 2, we present the expression of the solution and analyze the ill-posedness of problem (1.2). The a priori and a posteriori parameter choice rules which yield error estimates of Hölder type are suggested in Section 3. In Section 4, some numerical examples are given to illustrate the validity of the theoretical results. Finally, Section 5 ends this paper with a short conclusion.
2 Expression of the solution
Lemma 2.1
Proof
In order to get a better understanding of the property of solution (2.2), it is necessary to list some important properties of function \(\Phi(r,\xi)\). The following lemma establishes the relationship between \(\Phi(r,\xi)\) and some basic elementary functions.
Lemma 2.2
Proof
Case 1: \(|\xi|\geq1\).
Case 2: \(0<|\xi|<1\).
3 Fourier method and error estimates
Since the ill-posedness of problem (1.2) is caused by the high frequency perturbation of the noisy data, it is reasonable to stabilize the problem by eliminating high frequencies of the noisy data directly from the solution. This is the so-called Fourier method, it was put forward first by Lars Eldén et al. to deal with the inverse heat conduction problem [20]. Afterwards, this method has been successfully applied to deal with various inverse problems, e.g. the problem of a numerical pseudodifferential operator [21], the problem of numerical analytic continuation [22], the Cauchy problem for the Helmholtz equation [23], etc. However, for the Cauchy problem in a hollow cylinder, there are few efficient numerical methods, especially with a posteriori regularization parameter choice rule. In the following, we attempt to solve this problem by using Fourier method together with both a priori and a posteriori parameter choice rules.
3.1 A priori parameter choice rule
Theorem 3.1
Proof
3.2 A posteriori parameter choice rule
Lemma 3.1
- 1.
\(\rho(\xi_{\max})\) is a continuous and decreasing function on \((0,\infty)\),
- 2.
\(\lim_{\xi_{\max}\rightarrow\infty}\rho(\xi_{\max})=0\),
- 3.
\(\lim_{\xi_{\max}\rightarrow0}\rho(\xi_{\max})=\| g^{\delta}\|\).
For the choice rule of regularization parameter, we have a range estimate for \(\xi_{\max}\) given by the following lemma.
Lemma 3.2
Proof
Lemma 3.3
Proof
Lemma 3.4
Proof
Theorem 3.2
The proposition of Theorem 3.2 follows immediately from Lemmas 3.3 and 3.4.
4 Numerical experiment
Example 1
\(f(z)=e^{-z^{2}}\), \(z\in\mathbb {R}\).
Relative errors RES with \(\pmb{r=0.2}\) , \(\pmb{\varepsilon=0.001}\) , \(\pmb{M=31}\) for Example 1
K | 99 | 109 | 139 | 169 | 199 |
---|---|---|---|---|---|
RES (a priori) | 0.0038 | 0.0036 | 0.0036 | 0.0034 | 0.0035 |
RES (a posteriori) | 0.0037 | 0.0035 | 0.0036 | 0.0033 | 0.0038 |
Relative errors RES with \(\pmb{r=0.2}\) , \(\pmb{\varepsilon=0.001}\) , \(\pmb{K=109}\) for Example 1
M | 21 | 31 | 41 | 51 | 61 |
---|---|---|---|---|---|
RES (a priori) | 0.0036 | 0.0036 | 0.0036 | 0.0036 | 0.0035 |
RES (a posteriori) | 0.0035 | 0.0035 | 0.0034 | 0.0034 | 0.0034 |
Comparison of relative errors for Example 1
r | ε = 0.005 | ε = 0.05 | ||||
---|---|---|---|---|---|---|
0.2 | 0.6 | 0.9 | 0.2 | 0.6 | 0.9 | |
RES (a priori) | 0.0129 | 0.0588 | 0.1759 | 0.1191 | 0.1359 | 0.2093 |
RES (a posteriori) | 0.0117 | 0.0538 | 0.1362 | 0.1147 | 0.1228 | 0.1721 |
From Figures 1 and 2, and Table 3, we see that there is almost no difference for the a posteriori and a priori Fourier method with exact a priori bound E when r is relatively small. However, with the increase of r, the numerical effect of the a posteriori Fourier method is better than the a priori one. It is generally known that a priori bound E has a great influence on the accuracy of regularized solutions computed by a priori method, and a wrong a priori bound may lead to bad regularized solutions. This is just the weakness of the a priori parameter choice rule. The following example will also confirm this matter.
Example 2
\(f(z)=\sin\frac{\pi z}{10}\), \(z\in\mathbb{R}\).
Comparison of relative errors for Example 2
r | ε = 0.005 | ε = 0.05 | ||||
---|---|---|---|---|---|---|
0.2 | 0.6 | 0.9 | 0.2 | 0.6 | 0.9 | |
RES (a priori) | 0.0041 | 0.0147 | 0.0871 | 0.0376 | 0.0468 | 0.0949 |
RES (a posteriori) | 0.0040 | 0.0096 | 0.0350 | 0.0363 | 0.0363 | 0.0377 |
The impact of a priori bound E on the relative errors for Example 2 at \(\pmb{r=0.9}\) with \(\pmb{\varepsilon=10^{-2}}\)
E | 1 | 3.1463 | 5 | 7 | 9 |
---|---|---|---|---|---|
RES | 0.0344 | 0.0782 | 0.1265 | 0.1605 | 0.1605 |
For linear ill-posed problems defined on a ‘strip’ or ‘cylinder’ domain, the Fourier method is the most simple and a very effective regularization method. We repeated the computations of Examples 1 and 2 using the modified Tikhonov regularization method. The advantage of this method is that explicit error estimate for some specific problems could be obtained. The expression of the modified Tikhonov regularized solution and the corresponding error estimate are listed in the appendix.
5 Conclusion
In this paper, we have applied the Fourier method together with a priori parameter choice rule and a posteriori parameter choice rule to solve the Cauchy problem for the Laplace equation in a hollow cylinder domain. The Hölder type error estimates between the exact solution and its approximation are obtained. As for any a priori regularization method, the choice of the regularization parameter usually depends on both the a priori bound and the noise level. In general, the a priori bound cannot be known exactly in practice, and working with a wrong a priori bound may lead to bad regularization solution. The advantage of the a posteriori method is that one does not need to know the smoothness and the a priori bound of the unknown solution. The numerical results also show that the Fourier method with a posteriori parameter choice rule is much stable than the one with a priori parameter choice rule for larger r and ε. However for the a posteriori method, some important information as regards the solution is concealed and hidden for the discrepancy principle, such that the theoretical analysis of the convergence rate is rather difficult obtain for some problems. The related theory is particularly worthy of further development.
Declarations
Acknowledgements
The authors would like to thank the editor and the referees for their valuable comments and suggestions, which improved the quality of their paper. The work is supported by the National Natural Science Foundation (NNSF) of China (Nos. 11171136 and 11571295), the Tianyuan Fund for Mathematics of the NSF of China (No. 11326235), the Youth Foundation of NNSF of China (Nos. 11301301, 61403327 and 11401513), the Natural Science Foundation of Shandong Province (No. BS2013SF027), and the Foundation of Shandong Educational Committee (No. J12LI06).
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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