An Integral Equations Method for the Cauchy Problem Connected with the Helmholtz Equation

The Cauchy problem for the Helmholtz equation arises in many areas of science, such as wave propagation, vibration, and electromagnetic scattering [1–4]. It is well known that the Cauchy problem is unstable. The solution is unique in some proper solution spaces, but it does not depend continuously on the Cauchy data. For the stability of this problem, we can refer to [5–7]. There are many authors in the literature to investigate this problem, andmany numerical methods are proposed. In [8], Sun et al. investigate a potential function method for this method based on the Tikhonov regularization. In [4, 9], Marin et al. investigate the boundary elementmethod via alternating iterative and conjugate gradient method. The boundary knot method can be found by Jin and Zheng [10, 11]. For the method of fundamental solutions, we can refer toMarin and Lesnic [12] andWei et al. [13]. Study on themomentmethod andboundary particlemethod can be found in Wei et al. [14] and Chen and Fu [15]. The main purpose of this paper is to provide a numerical method for solving the Cauchy problem connected with the Helmholtz equation. The main idea is to formulate integral equations to the Cauchy problem by Green’s representation theorem for the solution of the Helmholtz equation. This method was used to reconstruct the shape for the Laplace equation, we refer to Cakoni et al. [16, 17], and to solve a Cauchy problem by Chapko and Johansson [18]. In [19], the authors gave a numerical method of the Cauchy problem for the Laplace equation by using single-layer potential function and jump relations and discussed the decay rate for singular values of Laplacian via singular value decomposition. The outline of this paper is as follows. In Section 2, we present the formulation of integral equations to the Cauchy problem. In Section 3, we solve the integral equations by the Tikhonov regularization method with the Morozov principle and analyze the convergence and stability. Finally, two numerical examples are included to show the effectiveness of our method.


Introduction
The Cauchy problem for the Helmholtz equation arises in many areas of science, such as wave propagation, vibration, and electromagnetic scattering [1][2][3][4].It is well known that the Cauchy problem is unstable.The solution is unique in some proper solution spaces, but it does not depend continuously on the Cauchy data.For the stability of this problem, we can refer to [5][6][7].There are many authors in the literature to investigate this problem, and many numerical methods are proposed.In [8], Sun et al. investigate a potential function method for this method based on the Tikhonov regularization.In [4,9], Marin et al. investigate the boundary element method via alternating iterative and conjugate gradient method.The boundary knot method can be found by Jin and Zheng [10,11].For the method of fundamental solutions, we can refer to Marin and Lesnic [12] and Wei et al. [13].Study on the moment method and boundary particle method can be found in Wei et al. [14] and Chen and Fu [15].
The main purpose of this paper is to provide a numerical method for solving the Cauchy problem connected with the Helmholtz equation.The main idea is to formulate integral equations to the Cauchy problem by Green's representation theorem for the solution of the Helmholtz equation.This method was used to reconstruct the shape for the Laplace equation, we refer to Cakoni et al. [16,17], and to solve a Cauchy problem by Chapko and Johansson [18].In [19], the authors gave a numerical method of the Cauchy problem for the Laplace equation by using single-layer potential function and jump relations and discussed the decay rate for singular values of Laplacian via singular value decomposition.
The outline of this paper is as follows.In Section 2, we present the formulation of integral equations to the Cauchy problem.In Section 3, we solve the integral equations by the Tikhonov regularization method with the Morozov principle and analyze the convergence and stability.Finally, two numerical examples are included to show the effectiveness of our method.

Formulation of Integral Equations
Let  ⊂ R 2 be a bounded and simply connected domain with a regular boundary  ∈ C 2 and let  consist of two nonintersecting parts Γ and Σ, Σ ∪ Γ = , where Γ and Σ are nonempty.In general, we assume that Γ is an open-connected subset of .Consider the following Cauchy problem.Given Cauchy data   and   on Γ, we find , such that  satisfies where  is the unit normal to the boundary  directed into the exterior of  and the wave number  > 0. Without loss of generality, we make the assumption on the measured data that   ∈  1 (Γ) and   ∈  2 (Γ) and suppose that the Cauchy problem has a unique solution  in  3/2 () [14,20].From Green's representation theorem for the solutions of the Helmholtz equation [21], we know that the solution  of (1) has the following form: Here, Φ(, ) = (/4) (1)  0 (| − |).From the jump relations, we have Then, we have the following integral equations: Theorem 1. Integral equation (5) has at most one solution.
Proof.It is sufficient to prove that the homogeneous problem has a unique solution (| Σ , (/])| Σ ) = (0, 0), which means that the following equations: have a unique solution (| Σ , (/])| Σ ) = (0, 0).Let By (6), we know that ()| Γ = 0. From the properties of single-double layer and the jump relations [22][23][24], we deduce By ( 7), we know that from the radiation at infinite and the uniqueness of the exterior boundary value problem for the Helmholtz equation yields that  vanishes in the exterior of .So  = 0 in  2 \ .Thus, we can easily get ∈ C 2 yields the uniqueness of the Cauchy problem [7], and we conclude that  = 0 in  2 /.From the jump relations [25], we have This completes the proof.
For simplicity, we define some operators and symbols as follows: By the above definitions, we have the following simple equations: Supposing that the endpoints of Γ are  and , we can find that V satisfies the Helmholtz equation and satisfies V() = (), V() = (); let () = () − V(); then () is a solution of the Helmholtz equation and () = () = 0, so we can fix   () =   () = 0 and define where Remark 2. For the construction of the function V, we can give a simple example.Supposing that  = (0, 0) and  = (1, 0), From zero extension, we will get the following lemma.Theorem 3.6]; thus, the operators  2 and  2 are compact from  2 (Σ) to  2 (Σ) and  1 and  1 are compact from  2 (Σ) to  2 (Γ).

Lemma 3. The operator
From Theorem 1, we know that the Cauchy problem has a unique solution without the restriction on  2 , and thus the homogeneous problem has only trivial solution.With the aid of the jump relations, it can be seen that  2 − (1/2) has a trivial null space (for details see [26,Chapter 3.4]).From the Rizes-Fredholm theorem, we can easily get the following theorem.
By the above conclusion, we can get following equations: To this end, we define the operator N :  2 (Σ) →  2 (Γ) by Then, the following property of the operator N holds.
Theorem 5.The operator N :  2 (Σ) →  2 (Γ) is compact and injective.Proof.By Lemma 3, we know the operator N is compact.By Theorems 1 and 4, we deduce that the operator N is injective.Now, we turn to introducing our numerical algorithm.First, function  is achieved by solving the following integral equation: where Remark 6.In general, (19) is not solvable since we cannot assume that the Cauchy data ℎ, especially the measured noisy data ℎ  , are in the range N( 2 (Γ)) of N. Therefore, we will solve (19) by some regularization methods in the next section and then give the error estimates.

Tikhonov Regularization and Morozov Discrepancy Principle
In this section, we will use the Tikhonov regularization method and the Morozov discrepancy principle to solve the integral system (19) and then give the error estimates and convergence results.In general, we give the noise data , and then we should consider the following equations: Here ℎ  ∈  2 (Γ) are measured noisy data satisfying and it is obvious that  = O( 1 ).
The Tikhonov regularization of integral system ( 21) is to solve the following equation: By introducing the regularization operators we can achieve the regularized solution    =   ℎ  of ( 21).We choose the regularization parameter  by the Morozov discrepancy principle, and then we have the following result.Theorem 7. Let  be sufficiently small positive constant and  < ‖ℎ  ‖  2 (Γ) .Let the Tikhonov solution   () satisfy Here  ∈  2 (Σ) is the exact solution which satisfies (19).
Proof.The statement follows directly from Theorem 2.17 in [25].
Consider the following Neumann boundary value problem: where  1 = O(), we know that there is a unique weak solution in  1 () [14].
Then we have the following main result in this paper.
Theorem 8. Let the assumptions in Theorem 7 hold.Then Moreover, the following estimate on boundary Σ holds: The positive constant  depends only on , , and .Proof.From triangle inequality and Theorem 7, we get The inequalities imply the estimate (28).
From the assumption, we have The trace theorem and the triangle inequality yield the estimate (27).

Numerical Examples
In this section, we report two examples of R 2 to test the effectiveness of our method.In the figures, we denote by  and  the function values and the normal derivative values, respectively.For the discrete of the integral equations, we use the Nyström method, see [23,Chapter 3.5].
Example 1.To test our code, consider the case in which the exact solution to the Cauchy problem is , and let Σ =  \ Γ.In this example, we observe the effect of noise on the numerical solution on Σ.
The regularization parameters  chosen by the Morozov discrepancy principle and the errors are given in Tables 1 and  2.
Figures 1, 2, 3, and 4 show the real part of the numerical solutions for different wave numbers with different levels of noise of Cases 1 and 2, respectively.
From the figures and tables, it can be seen that the numerical solutions are stable approximations of the exact solution, and it should be noted that the numerical solution   , where () is the polar angle of  and Θ is a specified angle.In this example, we observe the effect of Θ on the numerical solution.Choose () =  1 ()  as the exact solution, where  1 is the Bessel function of order one.Tables 3 and 4 give the regularization parameters and present the corresponding  2 errors and relative  2 errors for the approximation of  and / on boundary Σ.In order to investigate the effect of Θ, Figures 6 and 7 show the real part of the numerical solutions with different Θ.It can be seen that large Θ will improve the results.

Conclusions
In this paper, we study the application of an integral equations method to solve the Cauchy problem connected with the Helmholtz equation.We give the uniqueness of this problem in Theorem 1, in Section 2, and this cannot be obtained directly since the restriction on  2 .Then we use the Tikhonov

Figure 1 :
Figure 1: Example 1: the exact solution and the numerical solution on Σ with  = 3 for Case 1.

Figure 2 :
Figure 2: Example 1: the exact solution and the numerical solution on Σ with  = 8 for Case 1.

Figure 3 :
Figure 3: Example 1: the exact solution and the numerical on Σ with  = 1 for Case 2.

Figure 4 :
Figure 4: Example 1: the exact solution and the numerical solution on Σ with  = 3 for Case 2.

Figure 5 :
Figure 5: Example 2: the exact solution and the numerical solution on Σ with different noises,  = 5.

Figure 5
Figure5shows the real part of the numerical solution with different levels of noise on Θ = .In order to investigate the effect of Θ, Figures6 and 7show the real part of the numerical solutions with different Θ.It can be seen that large Θ will improve the results.

Table 2 :
Regularization parameter  and errors for Example 1 of Case 1 with  = 8.