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

Authors: Yao Sun; Deyue Zhang
Journal: Mathematical Problems in Engineering
 (2013)

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## 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.


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## Body

## 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 Section2, 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

LetD⊂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 fD and fN on Γ, we find u, such that u satisfies
(1)Δu+k2u=0,inD,(2)u=fD,onΓ,∂u∂n=fN,onΓ,
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 fD∈H1(Γ) and fN∈L2(Γ) and suppose that the Cauchy problem has a unique solution u in H3/2(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:
(3)u(x)=∫∂D‍{∂u∂ν(y)Φ(x,y)-u(y)Φ(x,y)∂ν(y)}ds(y),x∈D.Here,Φ(x,y)=(i/4)H0(1)(k|x-y|).From the jump relations, we have(4)12u(x)=∫∂D‍{∂u∂ν(y)Φ(x,y)-u(y)Φ(x,y)∂ν(y)}ds(y),x∈∂D.Then, we have the following integral equations:(5)∫Σ‍{∂u∂ν(y)Φ(x,y)-u(y)Φ(x,y)∂ν(y)}ds(y)=12u(x)-∫Γ‍{∂u∂ν(y)Φ(x,y)-u(y)Φ(x,y)∂ν(y)}ds(y),x∈Γ,∫Σ‍{∂u∂ν(y)Φ(x,y)-u(y)Φ(x,y)∂ν(y)}ds(y)-12u(x)=-∫Γ‍{∂u∂ν(y)Φ(x,y)-u(y)Φ(x,y)∂ν(y)}ds(y),x∈Σ.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:
(6)∫Σ‍{∂u∂ν(y)Φ(x,y)-u(y)Φ(x,y)∂ν(y)}ds(y)=0,x∈Γ,(7)∫Σ‍{∂u∂ν(y)Φ(x,y)-u(y)Φ(x,y)∂ν(y)}ds(y)-12u(x)=0,x∈Σ,
have a unique solution (u|Σ,(∂u/∂ν)|Σ)=(0,0). Let
(8)ω(x)=∫Σ‍{∂u∂ν(y)Φ(x,y)-u(y)Φ(x,y)∂ν(y)}ds(y),x∈R2∖∂D.
By (6), we know that ω(x)∣Γ=0. From the properties of single-double layer and the jump relations [22–24], we deduce
(9)limx→Σ+ω(x)=∫Σ‍{∂u∂ν(y)Φ(x,y)-u(y)Φ(x,y)∂ν(y)}ds(y)-12u(x).
By (7), we know that
(10)limx→Σ+ω(x)=0,
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
(11)ω=0,x∈Γ,∂ω∂n=0,x∈Γ.∂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
(12)u|Σ=u|Σ--u|Σ+=0,∂u∂ν|Σ=∂u∂ν|Σ+-∂u∂ν|Σ-=0.
This completes the proof.For simplicity, we define some operators and symbols as follows:(13)(A1φ)(x)=∫Σ‍φ(y)Φ(x,y)ds(y),x∈Γ,(B1φ)(x)=-∫Σ‍φ(y)Φ(x,y)∂ν(y)ds(y),x∈Γ,(A2φ)(x)=∫Σ‍φ(y)Φ(x,y)ds(y),x∈Σ,(B2φ)(x)=-∫Σ‍φ(y)Φ(x,y)∂ν(y)ds(y),x∈Σ,f(x)=12u(x)-∫Γ‍{∂u∂ν(y)Φ(x,y)-u(y)Φ(x,y)∂ν(y)}ds(y),x∈Γ,g(x)=-∫Γ‍{Φ(x,y)∂ν(y)∂u∂ν(y)Φ(x,y)-u(y)Φ(x,y)∂ν(y)}ds(y),x∈Σ,U(x)=u(x)|Σ,V(x)=∂u∂ν(x)|Σ.By the above definitions, we have the following simple equations:(14)(A1V)(x)+(B1U)(x)=f(x),x∈Γ,(A2V)(x)+((B2-12I)U)(x)=g(x),x∈Σ.Supposing that the endpoints ofΓ 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 fD(A)=fD(B)=0 and define
(15)(A2′ψ)(x)=∫∂Dψ(y)Φ(x,y)ds(y),x∈Σ,
where
(16)ψ(y)={φ(y),y∈Σ,0,y∈Γ.Remark 2.
For the construction of the functionv, we can give a simple example. Supposing that A=(0,0) and B=(1,0), uA=a, uB=b, a≠b, we can fix v(x)=a(1-x1)eikx2+bx2eikx2.From zero extension, we will get the following lemma.Lemma 3.
The operatorA2′ 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 Theorem1, 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 operatorB2-(1/2)I is bounded invertible.By the above conclusion, we can get following equations:(17)[A1-B1(B2-I2)-1A2]V=f-B1(B2-I2)-1g,x∈Γ,U=(B2-I2)-1(g-A2V),x∈Σ.To this end, we define the operator𝒩:L2(Σ)→L2(Γ) by
(18)𝒩φ(x)=[A1-B1(B2-I2)-1A2]φ(x).Then, the following property of the operator𝒩 holds.Theorem 5.
The operator𝒩:L2(Σ)→L2(Γ) is compact and injective.Proof.
By Lemma3, 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, functionϕ is achieved by solving the following integral equation:
(19)𝒩V=h(x),x∈Γ,
where
(20)h(x)=f-B1(B2-I2)-1g,x∈Γ.Remark 6.
In general, (19) is not solvable since we cannot assume that the Cauchy data h, especially the measured noisy data hδ, 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 fDδ1, fNδ1, and then we should consider the following equations:
(21)𝒩Vδ=hδ.Herehδ∈L2(Γ) are measured noisy data satisfying
(22)∥h-hδ∥L2(Γ)≤δ,
and it is obvious that δ=𝒪(δ1).The Tikhonov regularization of integral system (21) is to solve the following equation:
(23)αVαδ+𝒩*𝒩Vαδ=𝒩*hδ.By introducing the regularization operators(24)Rα:=(αI+𝒩*𝒩)-1𝒩*,forα>0,we can achieve the regularized solutionVαδ=Rαhδ of (21). We choose the regularization parameter α by the Morozov discrepancy principle, and then we have the following result.Theorem 7.
Letδ be sufficiently small positive constant and δ<∥hδ∥L2(Γ). Let the Tikhonov solution Vα(δ)δ satisfy ∥𝒩Vα(δ)δ-hδ∥L2(Γ)=δ for all δ∈(0,δ0) and let V=𝒩*z∈𝒩*(L2(Γ)) with ∥z∥L2(Γ)≤E. Then
(25)∥Vα(δ)δ-V∥L2(Σ)≤2δE.
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:(26)Δuα(δ)δ+k2uα(δ)δ=0,inD,∂uα(δ)δ∂n=fNδ1,onΓ,∂uα(δ)δ∂n=Vαδ,onΣ,
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 Theorem7 hold. Then
(27)∥uα(δ)δ-u∥H1(D)≤C1δ1/2.
Moreover, the following estimate on boundaryΣ holds:
(28)∥uα(δ)δ-u∥L2(Σ)+∥∂uα(δ)δ∂n-∂u∂n∥L2(Σ)≤Cδ1/2.
The positive constantC depends only on k, D, and E.Proof.
From triangle inequality and Theorem7, we get
(29)∥uα(δ)δ-u∥L2(Σ)+∥∂uα(δ)δ∂n-∂u∂n∥L2(Σ)=∥(B2-I2)-1[(gδ-A1Vα(δ)δ)-(g-A1V)]∥L2(Σ)+∥Vα(δ)δ-V∥L2(Σ)≤C3∥gδ-g∥L2(Σ)+C4∥Vα(δ)δ-V∥L2(Σ)≤Cδ1/2.
The inequalities imply the estimate (28).
From the assumption, we have(30)∥uα(δ)δ-u∥L2(Γ)+∥∂uα(δ)δ∂n-∂u∂n∥L2(Γ)≤2δ1≤C′δ1/2.
Then, we get(31)∥uα(δ)δ-u∥L2(∂D)+∥∂uα(δ)δ∂n-∂u∂n∥L2(∂D)≤C′′δ1/2.
The trace theorem and the triangle inequality yield the estimate (27).


## 4. Numerical Examples

In this section, we report two examples ofℝ2 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 isu(x)=eikx·d. Let D={(x1,x2)∣x12+x22<0.52}, let Γ={(x1,x2)∣x12+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 choosed=(0,1).Case 2.
We choosed=(2/2,2/2).The regularization parametersα chosen by the Morozov discrepancy principle and the errors are given in Tables 1 and 2.Table 1
Regularization parameterα and errors for Example 1 of Case 1 with k=3.
Noise
α
∥
U
α
(
δ
)
δ
-
u
∥
L
2
(
∑
‍
)
∥
V
α
(
δ
)
δ
-
∂
n
u
∥
L
2
(
∑
‍
)
/
∥
∂
n
u
∥
L
2
(
∑
‍
)
0
7.78
×
1
0
-
15
8.29
×
1
0
-
4
3.6
×
1
0
-
3
0.001
2.53
×
1
0
-
6
2.20
×
1
0
-
2
5.62
×
1
0
-
2
0.01
2.08
×
1
0
-
4
6.58
×
1
0
-
2
1.21
×
1
0
-
1
0.03
7.75
×
1
0
-
4
8.53
×
1
0
-
2
1.33
×
1
0
-
1Table 2
Regularization parameterα and errors for Example 1 of Case 1 with k=8.
Noise
α
∥
U
α
(
δ
)
δ
-
u
∥
L
2
(
∑
‍
)
∥
V
α
(
δ
)
δ
-
∂
n
u
∥
L
2
(
∑
)
/
∥
∂
n
u
∥
L
2
(
∑
‍
)
0
3.36
×
1
0
-
16
9
×
1
0
-
3
4.25
×
1
0
-
2
0.001
1.16
×
1
0
-
6
3.56
×
1
0
-
2
8.07
×
1
0
-
2
0.01
3.14
×
1
0
-
5
5.13
×
1
0
-
2
1.07
×
1
0
-
1
0.03
2.85
×
1
0
-
4
5.24
×
1
0
-
2
1.19
×
1
0
-
1Figures1, 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.Example1: the exact solution and the numerical solution on Σ with k=3 for Case 1.
(a)
f
(b)
dExample1: the exact solution and the numerical solution on Σ with k=8 for Case 1.
(a)
f
(b)
dExample1: the exact solution and the numerical solution on Σ with k=1 for Case 2.
(a)
f
(b)
dExample1: the exact solution and the numerical solution on Σ with k=3 for Case 2.
(a)
f
(b)
dFrom 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 discD={(x1,x2)∣x12+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)eiθ as the exact solution, where J1 is the Bessel function of order one.Tables3 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 parameterα and errors for Example 2 with k=5, 1% noise.
Θ
α
∥
U
α
(
δ
)
δ
-
u
∥
L
2
(
∑
‍
)
∥
V
α
(
δ
)
δ
-
∂
n
u
∥
L
2
(
∑
‍
)
/
∥
∂
n
u
∥
L
2
(
∑
‍
)
π
/
2
2.02
×
1
0
-
2
1.79
×
1
0
-
1
2.94
×
1
0
-
1
π
1.03
×
1
0
-
4
5.99
×
1
0
-
2
9.43
×
1
0
-
2
3
π
/
2
3.81
×
1
0
-
5
4.90
×
1
0
-
3
1.96
×
1
0
-
2Table 4
Regularization parameterα and errors for Example 2 with k=5, 3% noise.
Θ
α
∥
U
α
(
δ
)
δ
-
u
∥
L
2
(
∑
‍
)
∥
V
α
(
δ
)
δ
-
∂
n
u
∥
L
2
(
∑
‍
)
/
∥
∂
n
u
∥
L
2
(
∑
‍
)
π
/
2
4.04
×
1
0
-
2
2.32
×
1
0
-
1
3.13
×
1
0
-
1
π
3.13
×
1
0
-
4
7.37
×
1
0
-
2
1.19
×
1
0
-
1
3
π
/
2
1.54
×
1
0
-
4
1.83
×
1
0
-
2
5.79
×
1
0
-
2Figure5 shows the real part of the numerical solution with different levels of noise on Θ=π.Example2: the exact solution and the numerical solution on Σ with different noises, k=5.
(a)
f
(b)
dIn order to investigate the effect ofΘ, 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.Example2: the exact solution and the numerical solution on Σ with k=5, 1% noise.
(a)
f
(b)
dExample2: the exact solution and the numerical solution on Σ with k=5, 3% noise.
(a)
f
(b)
d


## 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 Theorem1, 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.



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