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Cauchy problem for the Laplace equation in a radially symmetric hollow cylinder
Boundary Value Problems volumeÂ 2016, ArticleÂ number:Â 195 (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 illposed 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.
1 Introduction
The Cauchy problem for the Laplace equation is an old yet persistent problem arising in many practical applications, and the general form is
where Î© is a domain in \(\mathbb{R}^{n}\), \(\Gamma\subset \partial\Omega\) is part of the boundary, g and h are given functions in \(L^{2}(\Gamma)\), and the solution u is sought in the whole domain Î©. This problem arises in many practical contexts, for example, in the problem of electrical prospecting, u denotes the potential of the electrostatic field artificially created in the interior of the Earth. We have \(\triangle u=0\), \(x\in\Omega\), where \(u_{\Gamma}\) is the magnitude of the potential u and \(\frac{\partial u}{\partial n}_{\Gamma}\) is the intensity of the potential, both measured at the accessible surface Î“ of the Earth. In general, this problem is illposed since for some Cauchy data g and h there is not a solution, and even if there exists a solution it does not always depend continuously on the data. Therefore, several regularization methods have been presented to solve it such as the quasireversibility method [1, 2], the boundary element method [3, 4], the Fourier regularization method [5, 6], the central difference regularization method [7], the mollification method [8], etc. However, most of the results are in two dimensions. For the high dimensional case, both theoretical analysis and numerical computation are very difficult. In [9], the authors transfer high dimensional Cauchy problem for Laplace equation into moment problem, and then construct a series of polynomial functions to approximate solutions of the moment problem. In [10], a quasiboundaryvalue method together with leftpreconditioned generalized minimum residual method are proposed to deal with an illposed Cauchy problem for a 3D elliptic partial differential equation with variable coefficients.
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]. Åžeremet and Åžeremet [16] presented new steadystate Greenâ€™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].
In this paper, suppose the considered 3D hollow cylinder domain is regular, and the internal and external radii are denoted \(r_{0}\) and R, respectively. Under the assumption that the inside surface of the cylinder is composed of insulating materials, it makes most sense to use cylindrical coordinates to transfer problem (1.1) into the following problem:
where \(r=\sqrt{x^{2}+y^{2}}\), and \(g(z)\) is a known potential distribution along the inside surface.
In practice, the data \(g(z)\) is often obtained by the instrument installed inside the hollow cylinder, and there exist unavoidable errors. We assume that instead of exact data \(g(z)\in L^{2}(\mathbb {R})\), only a noisy data \(g^{\delta}(z)\in L^{2}(\mathbb{R})\) with
is available. \(\delta>0\) represents the â€˜noise levelâ€™ and \(\\cdot\\) denotes the \(L^{2}\)norm.
We further assume that
and the following a priori bound holds:
where E is a positive constant.
This paper is organized as follows. In SectionÂ 2, we present the expression of the solution and analyze the illposedness of problem (1.2). The a priori and a posteriori parameter choice rules which yield error estimates of HÃ¶lder type are suggested in SectionÂ 3. In SectionÂ 4, some numerical examples are given to illustrate the validity of the theoretical results. Finally, SectionÂ 5 ends this paper with a short conclusion.
2 Expression of the solution
For \(f(z)\in L^{1}(\mathbb{R})\), \(\hat{f}(\xi)\) denotes its Fourier transform, which is defined by
Thus by using Fourier transform, the problem (1.2) is transformed into the following initial problem (2.1) of ODE in frequency domain:
Lemma 2.1
The solution of problem (2.1) is given by
where
and \(I_{0}(\cdot)\), \(I_{1}(\cdot)\), \(K_{0}(\cdot)\), \(K_{1}(\cdot)\) denote the modified Bessel function.
Proof
From [18], we know that the modified Bessel equation has two linearly independent solutions \(I_{0}\) and \(K_{0}\), then the general solution of equation in problem (2.1) is
Combining the boundary conditions in (2.1) with the properties \(I_{0}'(x)=I_{1}(x)\), \(K_{0}'(x)=K_{1}(x)\), and \(I_{0}(x)K_{1}(x)+I_{1}(x)K_{0}(x)=\frac{1}{x}\) for \(x>0\), we get equation (2.2) of the solution to problem (2.1).â€ƒâ–¡
Note that if \(r_{0}=0\), it is a special case to problem (1.2) and the corresponding solution is
There is only one modified Bessel function \(I_{0}\) in this expression, and this case has been discussed in [19].
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
For \(\xi\neq0\), there exist positive constants \(C_{1}\) and \(C_{2}\) such that the following inequalities hold:
Proof
Case 1: \(\xi\geq1\).
According to [18], the â€˜asymptotic expansions for large argumentsâ€™ of modified Bessel functions \(I_{0}\), \(I_{1}\), \(K_{0}\), and \(K_{1}\) are as follows:
and then, combining with the continuous property of Bessel functions, we know that there exist four pairs of positive constants \({c}_{\nu}\), \(c_{\nu}'\), \(d_{\nu}\), and \(d_{\nu}'\) (\(\nu=0,1\)), such that
Furthermore, on denoting \(\mu_{1}=c_{0}d_{1}\) and \(\mu _{2}=c_{0}'d_{1}'+c_{1}'d_{0}'\), then the following inequalities are straightforward calculations by using equation (2.3) and the above inequalities (2.6):
Case 2: \(0<\xi<1\).
Based on equation (2.3), we have
For ease of use, an alternative form of (2.8) is as follows:
and combining with the continuous property of \(\Phi(r,\xi)\) on \([r_{0},R]\times(0,1)\), we know that there exist two positive constants \(\mu_{3}\) and \(\mu_{4}\) such that the following inequalities hold:
If we take \(C_{1}=\min(\mu_{1},\mu_{3})\), and \(C_{2}=\max(\mu_{2},\mu_{4})\), then for \(\xi\neq0\) inequalities (2.4) are obtained.â€ƒâ–¡
By the Parseval equality, we know
Combining with (1.4), (2.9), and LemmaÂ 2.2, we know that the Fourier transform \(\hat{g}(\xi)\) of the exact data \(g(z)\) must decay rapidly for \(\xi\rightarrow\infty\) in order to ensure the convergence of (2.9). However, for the noisy data \(g^{\delta}(z)\), its Fourier transform may not possess such a property, and the noisy perturbation will be multiplied by the diverging factor \(\xi\Phi(r,\xi)\).
3 Fourier method and error estimates
Since the illposedness 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 socalled Fourier method, it was put forward first by Lars EldÃ©n 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.
According to [20], we eliminate all high frequencies from the solution and consider (2.2) only for \(\xi<\xi _{\max}\), i.e., define the Fourier regularization solution of problem (1.2) as
where \(\xi_{\max}>0\) is the regularization parameter to be determined, and Ï‡ is the characteristic function. In the following, we will establish error estimates between the exact solution and its regularization approximations.
3.1 A priori parameter choice rule
Theorem 3.1
Assume the conditions (1.3)(1.5) hold. The regularized solution of problem (1.2) is defined by (3.1). If we take the regularization parameter \(\xi_{\max}^{*}\) to be
then the following HÃ¶lder stability holds:
Proof
For the exact data \(g(z)\), we define
Combining with (2.2) and (3.1), we have
We will estimate \(I_{1}\) and \(I_{2}\) separately.
For the first term \(I_{1}\), due to (1.5), we have
For the second term \(I_{2}\), due to (1.3), we know that
Combining \(I_{1}\) and \(I_{2}\), we have
If we replace \(\xi_{\max}\) by \(\xi_{\max}^{*}\) defined by (3.2), the final estimate is obtained as (3.3). The proof is completed.â€ƒâ–¡
3.2 A posteriori parameter choice rule
Set
From equation (3.1) of the regularized solution \(v_{\xi_{\max}}^{\delta}\), it is easy to see that
Lemma 3.1
The function \(\rho(\xi_{\max})\) satisfies

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}\\).
According to the discrepancy principle, we will take the solution \(\xi _{\max}\) of the equation
to be the regularization parameter, where \(\tau>1\) is a constant. In practice, we always have \(\g^{\delta}\>\delta\), otherwise, \(v_{\xi _{\max}}^{\delta}\equiv0\) would be an acceptable approximation to u. Therefore, for an appropriate constant \(\tau>1\), equation (3.5) is always solvable, and if solution of (3.5) is not unique, the \(\xi_{\max}\) will be understood as the minimal solution of the equation.
For the choice rule of regularization parameter, we have a range estimate for \(\xi_{\max}\) given by the following lemma.
Lemma 3.2
Assume that conditions (1.3)(1.5) hold, if \(\xi_{\max}\) is taken as the solution of equation (3.5), then the following inequality holds:
Proof
It is easy to observe that
In view of equation (3.5) and the triangle inequality, we also have
Combining (3.7) with (3.8), we have
and therefore (3.6) holds.â€ƒâ–¡
Lemma 3.3
Assume that conditions (1.3)(1.5) hold, then we have
where C is a constant independent on Î´ and E.
Proof
Let the index \(\alpha=\frac{Rr}{Rr_{0}}\), it can be deduced from LemmaÂ 2.2 that there exists a positive constant C such that
Then we have the following estimate:
Thus the proof is completed.â€ƒâ–¡
Lemma 3.4
Assume that conditions (1.3)(1.5) hold. If \(\xi_{\max}^{*}\) is taken as the solution of equation (3.5), then we have
where \(C^{*}\) is a constant independent on Î´ and E.
Proof
From noise level (1.3) and the choice rule for \(\xi _{\max}^{*}\), the first inequality is easy to obtain,
For the second inequality, we have
Combining with LemmaÂ 3.2, we have
where \(C^{*}=1+\frac{C_{2}}{C_{1}(\tau1)}\). The proof of this lemma is completed.â€ƒâ–¡
Theorem 3.2
Assume that conditions (1.3)(1.5) hold. Further suppose that \(\delta<\g^{\delta}\\). Choose \(\tau>1\) such that \(0<\tau\delta<\g^{\delta}\\). If \(\xi_{\max}^{*}\) is taken as the solution of equation (3.5), then the following error estimates are satisfied:
where C and \(C^{*}\) are the same as in Lemmas 3.3 and 3.4, respectively.
The proposition of TheoremÂ 3.2 follows immediately from Lemmas 3.3 and 3.4.
4 Numerical experiment
In this section, we present the numerical implementation of the Fourier method with a priori and a posteriori parameter choice rule, respectively. The fast Fourier transform and the inverse fast Fourier transform are used to compute the approximate solutions, and all computations are done in Matlab 7.0. In the computation, we always take \(R=1\), \(r_{0}=0.1\), and we consider the problem in domain \(\{ 0.1< r<1, 10<z<10\}\). The numbers of grids on the \((r,z)\) domain are denoted M and K. In practical applications, the data \(g(z)\) is obtained by measurement and there are inevitable errors. Thus in our experiment, we will consider the noisy data \(g^{\delta}\) created by
where \(g=(g(z_{1}),g(z_{2}),\ldots,g(z_{K}))\), \(z_{k}=10+\frac {20(k1)}{K1}\), \(k=1,2,\ldots,K\), and the noise level
The function â€˜\(\operatorname{rand}(\cdot)\)â€™ generates arrays of uniformly distributed random numbers.
For the a priori parameter choice rule, the regularization parameter depends on both the a priori bound and the noise level. The a priori bound is computed according to
To test the accuracy of the computed approximations, we use the relative root mean square error (RES), which is defined as
We will consider two examples where there are no exact solutions. The data function \(g(z)\) is obtained by solving the following direct problem:
where \(f(z)\) is selected to be a function with some interesting features. The wellposed problem (4.2) could then be solved by the standard fivepoint difference scheme and the corresponding data function \(g(z)\) can be computed.
Example 1
\(f(z)=e^{z^{2}}\), \(z\in\mathbb {R}\).
TablesÂ 1 and 2 show the effect of increasing M and K on accuracy. From these tables, we find that M and K have small influence on the results when they become large. That is to say, the degree of illposedness of numerical problems does not increase with the refinement of the mesh used. Thus we shall always take \(M=31\), \(K=109\) in the numerical experiment.
TableÂ 3 lists the relative root mean square error (RES) between the exact solution and its approximations with different perturbations in the data. The approximations are obtained by the Fourier method with both the a priori and the a posteriori parameter choice rule. For the a priori Fourier method, the regularization parameter is selected according to TheoremÂ 3.1. For the a posteriori Fourier method, we choose \(\xi_{\max}^{*}\) as the solution of equation (3.5), where Ï„ is some constant greater than unity, which can be taken heuristically to be 1.1. Note that the regularization parameter chosen by the a posteriori rule is only dependent on noise level Î´, and computing accuracy is improved. This table shows that the smaller r, the better the computed results, which is consistent with the theoretical result in Theorems 3.1 and 3.2. The suggested method is still stable for higher noise levels on the data, and the smaller Îµ, the more accurate the approximations.
FigureÂ 1 is the comparison of exact solution and its approximations at different values of the radius \(r=0.2, 0.6, 0.9\). In FigureÂ 1(a1)(c1), the perturbation is \(\varepsilon=0.005\). In FigureÂ 1(a2)(c2), the perturbation is \(\varepsilon=0.05\).
FigureÂ 2 gives the corresponding comparison of exact solution and its approximations in terms of \(u(\cdot, z)\) distributions at different constant \(z=10, 0, 2, 5\) for \(\varepsilon=0.05\).
From FiguresÂ 1 and 2, and TableÂ 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
\(f(z)=\sin\frac{\pi z}{10}\), \(z\in\mathbb{R}\).
TableÂ 4 lists the RES between the exact solution and its approximations with different perturbations in the data for ExampleÂ 2.
FigureÂ 3 is a comparison of the exact solution and its approximations at different values of the radius \(r=0.2, 0.6, 0.9\) for ExampleÂ 2. In FigureÂ 3(a1)(c1), the perturbation is \(\varepsilon=0.005\). In FigureÂ 3(a2)(c2), the perturbation is \(\varepsilon=0.05\).
The explanation for TableÂ 4 and FigureÂ 3 is similar to that for ExampleÂ 1, but it is worth noting that numerical results for the a posteriori Fourier method are more accurate at \(r=0.9\). The reason for this phenomenon is mainly the that regularization parameter selected by a posteriori choice rule depends only on the noise level Î´, and is not related to other factors. TableÂ 5 gives a discussion as regards the impact of bound E on the relative error of the regularized approximation for ExampleÂ 2, and we also see that a wrong constant E may lead to bad regularized solutions.
For linear illposed problems defined on a â€˜stripâ€™ or â€˜cylinderâ€™ 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.
FigureÂ 4 shows the comparison between the a posteriori Fourier method and the modified Tikhonov method on \(r=0.9\) with different perturbations. From this figure, it is easy to see that the Fourier method is much stable and better for larger r and Îµ.
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 HÃ¶lder 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 Îµ. 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.
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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.Â 11171136 and 11571295), the Tianyuan Fund for Mathematics of the NSF of China (No.Â 11326235), the Youth Foundation of NNSF of China (Nos.Â 11301301, 61403327 and 11401513), the Natural Science Foundation of Shandong Province (No.Â BS2013SF027), and the Foundation of Shandong Educational Committee (No.Â J12LI06).
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Appendix
Appendix
In order to compare the results with Fourier method, we repeat the computations of Examples 1 and 2 using the modified Tikhonov regularization method. Define the approximation as follows:
Before giving the explicit error estimate, we present first the following lemma which is crucial for the error estimate.
Lemma
[24]
Let \(0< x< l\), then we have
Theorem
Assume that conditions (1.3)(1.5) hold. If the regularization parameter Î± is selected to be
then the HÃ¶lder type error estimate holds,
where \(C'\) is a constant independent on Î´ and E.
Proof
For fixed \(r\in(r_{0}, R)\),
Now we estimate \(I_{1}\) and \(I_{2}\) separately.
From the estimate of \(I_{1}\) and \(I_{2}\), we have
If we select \(\alpha=\frac{\delta}{E}\), then we have
where \(C'=C_{1}^{\frac{rr_{0}}{Rr_{0}}}C_{2}r_{0}^{\frac {Rr}{Rr_{0}}}+C_{1}^{\frac{2r_{0}Rr}{Rr_{0}}}C_{2}^{2}r_{0}^{2}\). The proof is completed.â€ƒâ–¡
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Ma, YJ., Fu, CL. Cauchy problem for the Laplace equation in a radially symmetric hollow cylinder. Bound Value Probl 2016, 195 (2016). https://doi.org/10.1186/s1366101607028
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DOI: https://doi.org/10.1186/s1366101607028
MSC
 35R25
 35R30
Keywords
 Cauchy problem for the Laplace equation
 hollow cylinder
 illposed problem
 regularization
 error estimates