Method of fundamental solutions for a Cauchy problem of the Laplace equation in a halfplane
Abstract
This paper is to provide an analysis of an illposed Cauchy problem in a halfplane. This problem is novel since the Cauchy data on the accessible boundary is given, whilst the additional temperature is involved on a line. The Dirichlet boundary condition on part of the boundary is an essential condition in the physical meaning. Then we use a redefined method of fundamental solutions (MFS) to determine the temperature and the normal heat flux on the inaccessible boundary. The present approach will give an illconditioned system, and this is a feature of the numerical method employed. In order to overcome the illposedness of the corresponding system, we use the Tikhonov regularization method combining Morozov’s discrepancy principle to get a stable solution. At last, four numerical examples, including a smooth boundary, a boundary with a corner, and a boundary with a jump, are given to show the effectiveness of the present approach.
Keywords
Cauchy problem Laplace equation Halfplane Single layer potential1 Introduction
Cauchy problems, which arise in diverse science areas such as wave propagation, nondestructive testing, and geophysics have been intensively studied in the past decades [1, 7, 13, 33]. On account of the incomplete boundary conditions, Cauchy problems are classified as inverse problems and illposed, i.e., the solutions do not depend continuously on the Cauchy data. In order to get a stable solution, various numerical methods have been proposed to solve Cauchy problems [23, 31, 36]. The method of fundamental solutions (MFS) is a popular and frequently used method for the solution of such problems.
The MFS is a meshless method which expands the solution utilizing fundamental solutions [8, 16, 19, 20, 22, 28, 30, 39, 40]. It is a boundary collocation method which belongs to the family of Trefftz methods, see [14] for a link to Trefftz methods, boundary element methods, and MFS. It is applicable to boundary value problems in which the fundamental solution of the operator in the governing equation is known. Since then, it has been successfully applied to a large variety of physical problems, an account of which can be found in the survey paper [16]. In [26], Liu et al. introduce a novel concept to construct Trefftz energy bases based on the MFS for the numerical solution of the Cauchy problem in an arbitrary star plane domain. The Trefftz energy bases used for the solution not only satisfy the Laplace equation but also preserve the energy. In [32], the authors give a meshless method based on the MFS for the threedimensional inverse heat conduction problems. In [27], Marin investigates both theoretically and numerically the socalled invariance property of the solution of boundary value problems associated with the anisotropic heat conduction equation in two dimensions. In [9], Fu et al. investigate the thermal behavior inside skin tissue with the presence of a tumor and use the method of approximate particular solutions to simulate a tumor in 3D. Following Fichera’s idea, Chen et al. [5] enriched the MFS by an added constant and a constraint. This enrichment condition ensures a unique solution of the problems considered. They also explained that this enrichment should be used when there is a degenerate scale. In [41], Zhang and Wei give two iterative methods for a Cauchy problem for an elliptic equation with variable coefficients in a strip region, the convergence rates of two algorithms are obtained by an apriori and an aposteriori selection rule for the regularization parameter. Other methods for the conduction problems can be found in [21, 37].
Cauchy problems have been investigated using the MFS because of the ease with which it can be implemented, particularly for the problems in complex geometries [10, 11, 25, 34]. Most of the studies consider Cauchy problems on the whole plane, but sometimes we should consider the problems in a halfplane [2, 3, 15, 24]. In [2], Chapko and Johansson consider a Cauchy problem for the Laplace equation in a twodimensional semiinfinite domain by a direct integral equation method. Later, they give a generalization of the situation to threedimensions with the Cauchy data only partially given. Compared with their previous work [2], they not only generalize that work to higher dimensions but also consider the more realistic case when the Cauchy data is only partially known, i.e., the Neumann/Dirichlet data measurements are specified throughout the plane bounding the upper halfspace, and the Dirichlet/Neumann data is given only on a finite portion of this plane.
The Cauchy problem in a halfplane is novel since the solution of the problem should satisfy a Dirichlet boundary condition on part of the boundary AB and the Cauchy data on the accessible boundary \(\varGamma _{1}\). We should note that a Dirichlet boundary condition on part of the boundary is an essential condition in the physical meaning. In this paper, for simplicity, we give a homogeneous boundary condition on the interface, i.e., the temperature is zero on the interface for a physical phenomenon.
In what follows, we describe a MFS for the numerical solution of the corresponding Cauchy problem. To prove the feasibility of the method, we use a single layer representation of the solution [18, 35]. Via the analysis of the single layer potentials, jump relations, and the Green’s function in the halfplane, the solution given by the MFS is proved to be an approximation of the genuine solution. An “auxiliary” curve is involved in the definition of the single layer potential to avoid singularity.
This paper is organized as follows. In Sect. 2, we describe the MFS in a halfplane and give some theoretical results for this method. In Sect. 3, we solve the equations by the Tikhonov regularization method with Morozov’s principle. Finally, four numerical examples, including a smooth boundary, a boundary with a corner, and a boundary with a jump, are presented to show the effectiveness of the presented method.
2 Formulation of the MFS
For theoretical analysis, the single layer potential representation is involved, which can be seen as a continuous version of the MFS.
Notice that we involve an “auxiliary” curve \(\varGamma'\) in (8) instead of directly defining the single layer potential on ∂D. Then singularities caused by the integral equation in the single layer potential are avoided since \(\varGamma'\) is apart from D̅. Another thing to notice is that the single layer potential (8) is related to the Green’s function in the halfplane instead of that in the free space. Further analysis is needed to get the properties of the single layer potential.
Denote by ν the unit outward normal vector to Λ, where \(\varOmega^{e}\) is the exterior of the boundary Λ. For a curve Γ and a function u, denote by \(\gamma_{\varGamma}^{+}u\) and \(\gamma_{\varGamma}^{}u\) the restrictions of u to Γ from exterior and interior, respectively. Denote by \(\partial_{\nu,\varGamma }^{+} u\) and \(\partial_{\nu,\varGamma}^{} u\) the normal derivatives on Γ from exterior and interior, respectively.
A symmetric continuation discussion shows that the Cauchy problem (1)–(3) is equivalent to the classical Cauchy problem with symmetric structure [3], which is well known to be uniquely solvable (for details, see [18]). Thus the Cauchy problem (1)–(3) also has a unique solution \(u\in H^{3/2}(D)\).
Lemma 1
Proof
Proposition 1
Proof
As we have discussed, \(u=S^{h}_{\varGamma'}\phi\) satisfies the Laplace equation (1) and the Dirichlet boundary condition \(u=0\).
Since \(\phi\in L^{2}(\varGamma')\) is the solution of the boundary integral equations (11)–(12), \(u\in H^{3/2}(\varOmega)\) satisfies the Cauchy boundary conditions (2)–(3) (for details of the spaces, see [18]). Thus \(u_{D}\) solves the Cauchy problem (1)–(3). The unique solvability of the Cauchy problem implies that the solution has representation (13).
This completes the proof. □
Finally, we give the following theorem for the MFS.
Theorem 1
Proof
3 Regularization method
 1.
Set \(n=0\), and give an initial regularization parameter \(\alpha_{0}>0\);
 2.
Get \(c_{\alpha_{n}}^{\delta}\) from \(({A}^{*}{A}+\alpha_{n} I)c_{\alpha_{n}}^{\delta}={A}^{*}b^{\delta}\);
 3.
Get \(\frac{\mathrm{d}}{\mathrm {d}\alpha}c^{\delta}_{\alpha_{n}}\) from \((\alpha_{n}I+{A}^{*}{A})\frac{\mathrm {d}}{\mathrm{d}\alpha}c^{\delta}_{\alpha_{n}}=c^{\delta}_{\alpha_{n}}\);
 4.Get \(F(\alpha_{n})\) and \(F'(\alpha_{n})\) byand$$F(\alpha_{n})= \bigl\Vert {A}c_{\alpha_{n}}^{\delta}b^{\delta} \bigr\Vert ^{2}\delta^{2} $$respectively.$$F'(\alpha_{n})=2\alpha_{n} \biggl\Vert {A} \frac{\mathrm{d}}{\mathrm{d}\alpha}c^{\delta}_{\alpha_{n}} \biggr\Vert ^{2} +2 \alpha_{n}^{2} \biggl\Vert \frac{{\mathrm{d}}}{{\mathrm{d}}\alpha}c^{\delta}_{\alpha_{n}} \biggr\Vert ^{2}, $$
 5.
Set \(\alpha_{n+1}=\alpha_{n}\frac{F(\alpha_{n})}{F'(\alpha_{n})}\). If \(\Vert {\alpha_{n+1}\alpha_{n}} \Vert <\varepsilon\) (\(\varepsilon\ll1\)), end. Else, set \(n=n+1\) and return to 2.
When the regularization parameter \(\alpha^{*}\) is fixed, we can obtain the regularized solution.
4 Numerical experiments
In this section, we provide some numerical examples to show the effectiveness of the proposed method.
Example 1

Case 1: Γ is chosen as \(\boldsymbol{x}_{1}=(0.6+0.1\cos 3t)(\cos t,\sin t)\), \(t\in[0,\pi]\).

Case 2: Γ is chosen as \(\boldsymbol{x}_{2}=(\cos t+0.65\cos 2t0.65, 1.5\sin t)\), \(t\in[0,\pi]\).

Case 3: Γ is chosen as \(\boldsymbol{x}_{3}=0.6\sqrt {4.25+2\cos3t}(\cos t,\sin t)\), \(t\in[0,\pi]\).
Example 2
Figure 5 shows the numerical solutions with different noise levels \(\delta=0.01\), \(\delta=0.03\), and \(\delta=0.05\). As is shown in Fig. 5, the computation of \(\frac{\partial u}{\partial\nu }\) is more sensitive to the error in this case.
Example 3
Example 4
5 Conclusion
In this paper, we have dealt with a Cauchy problem connected with the Laplace equation in a halfplane. With the Green’s function of the Laplacian in the halfplane, we have proposed a method of fundamental solutions to solve the Cauchy problem. Numerical experiments have also been given to show the effectiveness of the algorithm.
Notes
Acknowledgements
Not applicable.
Availability of data and materials
Not applicable.
Authors’ contributions
All of the authors contributed equally in writing this paper. All authors read and approved the final manuscript.
Funding
This work is supported by the Youth Key Teacher Program of Civil Aviation University of China and the National Natural Science Foundation of China (Nos. U1433202, 11501566) and the Scientific Research Foundation of Civil Aviation University of China (No. 2017QD04S).
Competing interests
All the authors declare that they have no competing interests.
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