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

Abstract

We are concerned with the Cauchy problem connected with the Helmholtz equation. We propose a numerical method, which is based on the Helmholtz representation, for obtaining an approximate solution to the problem, and then we analyze the convergence and stability with a suitable choice of regularization method. Numerical experiments are also presented to show the effectiveness of our method.

1. Introduction

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, 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.

2. Formulation of Integral Equations

Let 𝐷⊂ℝ² be a bounded and simply connected domain with a regular boundary 𝜕𝐷∈𝒞² and let 𝜕𝐷 consist of two nonintersecting parts Γ and Σ, Σ∪Γ=𝜕𝐷, where Γ and Σ are nonempty. In general, we assume that Γ is an open-connected subset of 𝜕𝐷. Consider the following Cauchy problem. Given Cauchy data 𝑓_𝐷 and 𝑓_𝑁 on Γ, we find 𝑢, such that 𝑢 satisfies Δ𝑢+𝑘²𝑢=0,in𝐷,𝑢=𝑓_𝐷,onΓ,𝜕𝑢𝜕𝑛=𝑓_𝑁,onΓ, (1)(2) where 𝑛 is the unit normal to the boundary 𝜕𝐷 directed into the exterior of 𝐷 and the wave number 𝑘>0. Without loss of generality, we make the assumption on the measured data that 𝑓_𝐷∈𝐻¹(Γ) and 𝑓_𝑁∈𝐿²(Γ) and suppose that the Cauchy problem has a unique solution 𝑢 in 𝐻^3/2(𝐷) [14, 20].

From Green’s representation theorem for the solutions of the Helmholtz equation [21], we know that the solution 𝑢 of (1) has the following form: 𝑢(𝑥)=_𝜕𝐷𝜕𝑢𝑦Φ𝑦Φ𝜕𝜈𝑥,𝑦−𝑢𝑥,𝑦𝑦𝑦,𝜕𝜈𝑑𝑠𝑥∈𝐷. (3)

Here, Φ(𝑥,𝑦)=(𝑖/4)𝐻₀⁽¹⁾(𝑘|𝑥−𝑦|).

From the jump relations, we have 12𝑢(𝑥)=_𝜕𝐷𝜕𝑢𝑦Φ𝑦Φ𝜕𝜈𝑥,𝑦−𝑢𝑥,𝑦𝑦𝑦,𝜕𝜈𝑑𝑠𝑥∈𝜕𝐷. (4)

Then, we have the following integral equations: _Σ𝜕𝑢𝑦Φ𝑦Φ𝜕𝜈𝑥,𝑦−𝑢𝑥,𝑦𝑦𝑦=1𝜕𝜈𝑑𝑠2𝑢(𝑥)−_Γ𝜕𝑢𝑦Φ𝑦Φ𝜕𝜈𝑥,𝑦−𝑢𝑥,𝑦𝑦𝑦,𝜕𝜈𝑑𝑠𝑥∈Γ,_Σ𝜕𝑢𝑦Φ𝑦Φ𝜕𝜈𝑥,𝑦−𝑢𝑥,𝑦𝑦𝑦−1𝜕𝜈𝑑𝑠2𝑢(𝑥)=−_Γ𝜕𝑢𝑦Φ𝑦Φ𝜕𝜈𝑥,𝑦−𝑢𝑥,𝑦𝑦𝑦,𝜕𝜈𝑑𝑠𝑥∈Σ. (5)

Theorem 1. Integral equation (5) has at most one solution.

Proof. It is sufficient to prove that the homogeneous problem has a unique solution (𝑢|_Σ,(𝜕𝑢/𝜕𝜈)|_Σ)=(0,0), which means that the following equations: _Σ𝜕𝑢𝑦Φ𝑦Φ𝜕𝜈𝑥,𝑦−𝑢𝑥,𝑦𝑦𝑦𝜕𝜈𝑑𝑠=0,𝑥∈Γ,_Σ𝜕𝑢𝑦Φ𝑦Φ𝜕𝜈𝑥,𝑦−𝑢𝑥,𝑦𝑦𝑦−1𝜕𝜈𝑑𝑠2𝑢(𝑥)=0,𝑥∈Σ, (6)(7) have a unique solution (𝑢|_Σ,(𝜕𝑢/𝜕𝜈)|_Σ)=(0,0). Let 𝜔(𝑥)=_Σ𝜕𝑢𝑦Φ𝑦Φ𝜕𝜈𝑥,𝑦−𝑢𝑥,𝑦𝑦𝑦,𝜕𝜈𝑑𝑠𝑥∈𝑅²⧵𝜕𝐷. (8) By (6), we know that 𝜔(𝑥)∣_Γ=0. From the properties of single-double layer and the jump relations [22–24], we deduce lim_𝑥→Σ+𝜔(𝑥)=_Σ𝜕𝑢𝑦Φ𝑦Φ𝜕𝜈𝑥,𝑦−𝑢𝑥,𝑦𝑦𝑦−1𝜕𝜈𝑑𝑠2𝑢(𝑥). (9) By (7), we know that lim_𝑥→Σ+𝜔(𝑥)=0, (10) from the radiation at infinite and the uniqueness of the exterior boundary value problem for the Helmholtz equation yields that 𝜔 vanishes in the exterior of 𝐷. So 𝜔=0 in 𝑅²⧵𝐷. Thus, we can easily get 𝜔=0,𝑥∈Γ,𝜕𝜔𝜕𝑛=0,𝑥∈Γ. (11)𝜕𝐷∈𝒞² yields the uniqueness of the Cauchy problem [7], and we conclude that 𝜔=0 in 𝑅²/𝜕𝐷. From the jump relations [25], we have 𝑢|_Σ=𝑢|_Σ⁻−𝑢|_Σ⁺=0,𝜕𝑢𝜕𝜈_Σ=𝜕𝑢𝜕𝜈_Σ⁺−𝜕𝑢𝜕𝜈_Σ⁻=0. (12) This completes the proof.

For simplicity, we define some operators and symbols as follows: 𝐴₁𝜑(𝑥)=_Σ𝜑𝑦Φ𝑦𝐵𝑥,𝑦𝑑𝑠,𝑥∈Γ,₁𝜑(𝑥)=−_Σ𝜑𝑦Φ𝑥,𝑦𝑦𝑦𝐴𝜕𝜈𝑑𝑠,𝑥∈Γ,₂𝜑(𝑥)=_Σ𝜑𝑦Φ𝑦𝐵𝑥,𝑦𝑑𝑠,𝑥∈Σ,₂𝜑(𝑥)=−_Σ𝜑𝑦Φ𝑥,𝑦𝑦𝑦𝑓(𝑥)=1𝜕𝜈𝑑𝑠,𝑥∈Σ,2𝑢(𝑥)−_Γ𝜕𝑢𝑦Φ𝑦Φ𝜕𝜈𝑥,𝑦−𝑢𝑥,𝑦𝑦𝑦,𝑔(𝑥)𝜕𝜈𝑑𝑠𝑥∈Γ,=−_Γ𝜕𝑢𝑦Φ𝑦Φ𝜕𝜈𝑥,𝑦−𝑢𝑥,𝑦𝑦𝑦,𝑈(𝑥)(𝑥𝜕𝜈𝑑𝑠𝑥∈Σ,=𝑢)|_Σ,𝑉(𝑥)=𝜕𝑢(𝑥𝜕𝜈)_Σ. (13)

By the above definitions, we have the following simple equations: 𝐴₁𝑉(𝑥)+𝐵₁𝑈(𝑥)(𝑥)𝐴=𝑓,𝑥∈Γ,₂𝑉(𝑥)+𝐵₂−12𝐼𝑈(𝑥)=𝑔(𝑥),𝑥∈Σ. (14)

Supposing that the endpoints of Γ are 𝐴 and 𝐵, we can find that 𝑣 satisfies the Helmholtz equation and satisfies 𝑣(𝐴)=𝑢(𝐴), 𝑣(𝐵)=𝑢(𝐵); let 𝜔(𝑥)=𝑢(𝑥)−𝑣(𝑥); then 𝜔(𝑥) is a solution of the Helmholtz equation and 𝜔(𝐴)=𝜔(𝐵)=0, so we can fix 𝑓_𝐷(𝐴)=𝑓_𝐷(𝐵)=0 and define 𝐴^′₂𝜓(𝑥)=_𝜕𝐷𝜓𝑦Φ𝑥,𝑦𝑑𝑠𝑦,𝑥∈Σ, (15) where 𝜓𝑦=𝜑𝑦,𝑦∈Σ,0,𝑦∈Γ. (16)

Remark 2. For the construction of the function 𝑣, we can give a simple example. Supposing that 𝐴=(0,0) and 𝐵=(1,0), 𝑢_𝐴=𝑎, 𝑢_𝐵=𝑏, 𝑎≠𝑏, we can fix 𝑣(𝑥)=𝑎(1−𝑥₁)𝑒^{𝑖𝑘𝑥2}+𝑏𝑥₂𝑒^{𝑖𝑘𝑥2}.

From zero extension, we will get the following lemma.

Lemma 3. The operator 𝐴^′₂ is compact from 𝐿²(𝜕𝐷) to 𝐻¹(𝜕𝐷) [21, Theorem 3.6]; thus, the operators 𝐴₂ and 𝐵₂ are compact from 𝐿²(Σ) to 𝐿²(Σ) and 𝐴₁ and 𝐵₁ are compact from 𝐿²(Σ) to 𝐿²(Γ).

From Theorem 1, we know that the Cauchy problem has a unique solution without the restriction on 𝑘², and thus the homogeneous problem has only trivial solution. With the aid of the jump relations, it can be seen that 𝐵₂−(1/2)𝐼 has a trivial null space (for details see [26, Chapter 3.4]). From the Rizes-Fredholm theorem, we can easily get the following theorem.

Theorem 4. The operator 𝐵₂−(1/2)𝐼 is bounded invertible.

By the above conclusion, we can get following equations: 𝐴₁−𝐵₁𝐵₂−𝐼2⁻¹𝐴₂𝑉=𝑓−𝐵₁𝐵₂−𝐼2⁻¹𝐵𝑔,𝑥∈Γ,𝑈=₂−𝐼2⁻¹𝑔−𝐴₂𝑉,𝑥∈Σ. (17)

To this end, we define the operator 𝒩∶𝐿²(Σ)→𝐿²(Γ) by 𝒩𝜑(𝑥)=𝐴₁−𝐵₁𝐵₂−𝐼2⁻¹𝐴₂𝜑(𝑥). (18)

Then, the following property of the operator 𝒩 holds.

Theorem 5. The operator 𝒩∶𝐿²(Σ)→𝐿²(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 𝜙 is achieved by solving the following integral equation: 𝒩𝑉=ℎ(𝑥),𝑥∈Γ, (19) where ℎ(𝑥)=𝑓−𝐵₁𝐵₂−𝐼2⁻¹𝑔,𝑥∈Γ. (20)

Remark 6. In general, (19) is not solvable since we cannot assume that the Cauchy data ℎ, especially the measured noisy data ℎ^𝛿, are in the range 𝒩(𝐿²(Γ)) of 𝒩. Therefore, we will solve (19) by some regularization methods in the next section and then give the error estimates.

3. 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 𝑓^𝛿1_𝐷, 𝑓^𝛿1_𝑁, and then we should consider the following equations: 𝒩𝑉^𝛿=ℎ^𝛿. (21)

Here ℎ^𝛿∈𝐿²(Γ) are measured noisy data satisfying ℎ−ℎ^𝛿_𝐿2(Γ)≤𝛿, (22) and it is obvious that 𝛿=𝒪(𝛿₁).

The Tikhonov regularization of integral system (21) is to solve the following equation: 𝛼𝑉^𝛿_𝛼+𝒩^∗𝒩𝑉^𝛿_𝛼=𝒩^∗ℎ^𝛿. (23)

By introducing the regularization operators 𝑅_𝛼∶=𝛼𝐼+𝒩^∗𝒩⁻¹𝒩^∗,for𝛼>0, (24)

we can achieve the regularized solution 𝑉^𝛿_𝛼=𝑅_𝛼ℎ^𝛿 of (21). We choose the regularization parameter 𝛼 by the Morozov discrepancy principle, and then we have the following result.

Theorem 7. Let 𝛿 be sufficiently small positive constant and 𝛿<‖ℎ^𝛿‖_𝐿²_(Γ). Let the Tikhonov solution 𝑉^𝛿_𝛼(𝛿) satisfy ‖𝒩𝑉^𝛿_𝛼(𝛿)−ℎ^𝛿‖_𝐿2(Γ)=𝛿 for all 𝛿∈(0,𝛿₀) and let 𝑉=𝒩^∗𝑧∈𝒩^∗(𝐿²(Γ)) with ‖𝑧‖_𝐿²_(Γ)≤𝐸. Then 𝑉^𝛿_𝛼(𝛿)−𝑉_𝐿2(Σ)≤2√𝛿𝐸. (25) Here 𝑉∈𝐿²(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: Δ𝑢^𝛿_𝛼(𝛿)+𝑘²𝑢^𝛿_𝛼(𝛿)=0,in𝐷,𝜕𝑢^𝛿_𝛼(𝛿)𝜕𝑛=𝑓^𝛿1_𝑁,onΓ,𝜕𝑢^𝛿_𝛼(𝛿)𝜕𝑛=𝑉^𝛿_𝛼,onΣ, (26) where 𝛿₁=𝒪(𝛿), we know that there is a unique weak solution in 𝐻¹(𝐷) [14].

Then we have the following main result in this paper.

Theorem 8. Let the assumptions in Theorem 7 hold. Then 𝑢^𝛿_𝛼(𝛿)−𝑢_𝐻1(𝐷)≤𝐶₁𝛿^1/2. (27)
Moreover, the following estimate on boundary Σ holds: 𝑢^𝛿_𝛼(𝛿)−𝑢_𝐿2(Σ)+𝜕𝑢^𝛿_𝛼(𝛿)−𝜕𝑛𝜕𝑢𝜕𝑛_𝐿2(Σ)≤𝐶𝛿^1/2. (28)
The positive constant 𝐶 depends only on 𝑘, 𝐷, and 𝐸.

Proof. From triangle inequality and Theorem 7, we get 𝑢^𝛿_𝛼(𝛿)−𝑢_𝐿2(Σ)+𝜕𝑢^𝛿_𝛼(𝛿)−𝜕𝑛𝜕𝑢𝜕𝑛_𝐿2(Σ)=𝐵₂−𝐼2⁻¹𝑔^𝛿−𝐴₁𝑉^𝛿_𝛼(𝛿)−𝑔−𝐴₁𝑉_𝐿2(Σ)+𝑉^𝛿_𝛼(𝛿)−𝑉_𝐿2(Σ)≤𝐶₃𝑔^𝛿−𝑔_𝐿2(Σ)+𝐶₄𝑉^𝛿_𝛼(𝛿)−𝑉_𝐿2(Σ)≤𝐶𝛿^1/2. (29)
The inequalities imply the estimate (28).
From the assumption, we have 𝑢^𝛿_𝛼(𝛿)−𝑢_𝐿2(Γ)+𝜕𝑢^𝛿_𝛼(𝛿)−𝜕𝑛𝜕𝑢𝜕𝑛_𝐿2(Γ)≤2𝛿₁≤𝐶^′𝛿^1/2. (30)
Then, we get 𝑢^𝛿_𝛼(𝛿)−𝑢_{𝐿2(𝜕𝐷)}+𝜕𝑢^𝛿_𝛼(𝛿)−𝜕𝑛𝜕𝑢𝜕𝑛_{𝐿2(𝜕𝐷)}≤𝐶^′′𝛿^1/2. (31)
The trace theorem and the triangle inequality yield the estimate (27).

4. Numerical Examples

In this section, we report two examples of ℝ² to test the effectiveness of our method. In the figures, we denote by 𝑓 and 𝑑 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 𝑢(𝑥)=𝑒^{𝑖𝑘𝑥⋅𝑑}. Let 𝐷={(𝑥₁,𝑥₂)∣𝑥²₁+𝑥²₂<0.5²}, let Γ={(𝑥₁,𝑥₂)∣𝑥²₁+𝑥²₂=0.5²,𝑥₂≥0}, and let Σ=𝜕𝐷⧵Γ. In this example, we observe the effect of noise on the numerical solution on Σ.

Case 1. We choose 𝑑=(0,1).

Case 2. We choose 𝑑=(√2/2,√2/2).

The regularization parameters 𝛼 chosen by the Morozov discrepancy principle and the errors are given in Tables 1 and 2.

Table 1 

Regularization parameter 𝛼 and errors for Example 1 of Case 1 with 𝑘=3.

Table 2 

Regularization parameter 𝛼 and errors for Example 1 of Case 1 with 𝑘=8.

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.

(a) 𝑓

(b) 𝑑

Figure 1 

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

(a) 𝑓

(b) 𝑑

Figure 2 

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

(a) 𝑓

(b) 𝑑

Figure 3 

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

(a) 𝑓

(b) 𝑑

Figure 4 

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

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 converges to the exact solution as the level of noise decreases.

Example 2. Consider the unit disc 𝐷={(𝑥₁,𝑥₂)∣𝑥²₁+𝑥²₂<1}. Let Γ={𝑥∈𝜕𝐷∣0<𝜃(𝑥)<Θ} and let Σ=𝜕𝐷⧵Γ={𝑥∈𝜕𝐷∣Θ<𝜃(𝑥)<2𝜋}, where 𝜃(𝑥) is the polar angle of 𝑥 and Θ is a specified angle. In this example, we observe the effect of Θ on the numerical solution. Choose 𝑢(𝑥)=𝐽₁(𝑘𝑟)𝑒^𝑖𝜃 as the exact solution, where 𝐽₁ is the Bessel function of order one.

Tables 3 and 4 give the regularization parameters and present the corresponding 𝐿² errors and relative 𝐿² errors for the approximation of 𝑢 and 𝜕𝑢/𝜕𝑛 on boundary Σ.

Table 3 

Regularization parameter 𝛼 and errors for Example 2 with 𝑘=5, 1% noise.

Table 4 

Regularization parameter 𝛼 and errors for Example 2 with 𝑘=5, 3% noise.

Figure 5 shows the real part of the numerical solution with different levels of noise on Θ=𝜋.

(a) 𝑓

(b) 𝑑

Figure 5 

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

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.

(a) 𝑓

(b) 𝑑

Figure 6 

Example 2: the exact solution and the numerical solution on Σ with 𝑘=5, 1% noise.

(a) 𝑓

(b) 𝑑

Figure 7 

Example 2: the exact solution and the numerical solution on Σ with 𝑘=5, 3% noise.

5. 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 𝑘². Then we use the Tikhonov regularization method with the Morozov discrepancy principle for solving this ill-posed problem. Convergence and stability of the method are then given with two examples. From the examples, we can see that the proposed method is more stable with more Cauchy data, and the numerical results are sensitive about the wavenumber.

Acknowledgments

The authors would like to thank the editors and the referee for their careful reading and valuable comments which lead to the improvement of the quality of the submitted paper. The research was supported by the NSFC (no. 11201476).