International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

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Volume 2012 |Article ID 435468 | https://doi.org/10.5402/2012/435468

Fang-Fang Dou, Chu-Li Fu, "A Wavelet Method for the Cauchy Problem for the Helmholtz Equation", International Scholarly Research Notices, vol. 2012, Article ID 435468, 18 pages, 2012. https://doi.org/10.5402/2012/435468

A Wavelet Method for the Cauchy Problem for the Helmholtz Equation

Academic Editor: L. Marin
Received16 Aug 2011
Accepted28 Sep 2011
Published04 Jan 2012

Abstract

We consider a Cauchy problem for the Helmholtz equation at a fixed frequency. The problem is severely ill posed in the sense that the solution (if it exists) does not depend continuously on the data. We present a wavelet method to stabilize the problem. Some error estimates between the exact solution and its approximation are given, and numerical tests verify the efficiency and accuracy of the proposed method.

1. Introduction

The Helmholtz equation is often used to approximate model wave propagation in inhomogeneous media. The demand for reliable numerical solutions to such type of problems is frequently encountered in geophysical and optoelectronic applications [1, 2]. In geophysical applications, for example, wave propagation simulations are used for the development of acoustic imaging techniques for gaining knowledge about geophysical structures deep within the Earth’s subsurface [3]. In optoelectronics, the determination of a radiation field surrounding a source of radiation (e.g., a light emitting diode) is also a frequently occurring problem [4]. In many engineering problems, the boundary conditions are often incomplete, either in the form of underspecified and overspecified boundary conditions on different parts of the boundary or the solution is prescribed at some internal points in the domain. These so-called Cauchy problems are inverse problems, and it is well known that they are generally ill posed in the sense of Hadamard [5]. However, the Cauchy problem suffers from the nonexistence and instability of the solution.

In this paper we consider the Cauchy problem for the Helmholtz equation in a “strip” 0<𝑥<1 as follows:Δ𝑢(𝑥,𝑦)+𝑘2𝑢(𝑥,𝑦)=0,𝑥(0,1),𝑦𝑛,𝑛1,𝑢(0,𝑦)=𝑔(𝑦),𝑦𝑛,𝑢𝑥(0,𝑦)=0,𝑦𝑛,(1.1) where Δ=𝜕2/𝜕𝑥2+𝑛𝑖=1𝜕2/𝜕𝑦2𝑖 is an 𝑛+1 dimensional Laplace operator. We want to determine the solution 𝑢(𝑥,𝑦) for 0<𝑥1 from the data 𝑔(𝑦). Due to the importance of its application, this problem has been studied by many researchers, for example, DeLillo et al. [6, 7], Jin and Zheng [8], Johansson and Martin [9], and Marin et al. [1014].

Let 𝒮 be the Schwartz space over 𝑛, and let 𝒮 be its dual (the space of tempered distributions). Let 𝑓 denote the Fourier transform of function 𝑓(𝑦)𝒮 defined by1𝑓(𝜉)=(2𝜋)𝑛/2𝑛𝑒𝑖𝜉𝑦𝜉𝑓(𝑦)𝑑𝑦,𝜉=1,,𝜉𝑛𝑦,𝑦=1,,𝑦𝑛,(1.2) while the Fourier transform of a tempered distribution 𝑓𝒮 is defined by=𝜙𝑓,𝜙𝑓,,𝜙𝒮.(1.3) In this paper, we will consider functions depending on the variables 𝑥[0,1], 𝑦𝑛.

For 𝑠, the Sobolev space 𝐻𝑠(𝑛) consists of all tempered distributions 𝑓(𝑦)𝒮, for which 𝑓(𝜉)(1+|𝜉|2)𝑠/2 is a function in 𝐿2(𝑛). The norm on this space is given by𝑓𝐻𝑠=𝑛||||𝑓(𝜉)2||𝜉||(1+2)𝑠𝑑𝜉1/2.(1.4)

We assume there exists a unique solution 𝑢(𝑥,𝑦) of problem (1.1), which satisfies the problem in the classical sense and 𝑔(), 𝑢(𝑥,)𝐿2(𝑛). Applying the Fourier transform technique to problem (1.1) with respect to the variable 𝑦 yields the following problem in the frequency space:̂𝑢𝑥𝑥𝑘(𝑥,𝜉)+2||𝜉||2̂𝑢(𝑥,𝜉)=0,𝑥(0,1),𝜉𝑛,𝑛1,̂𝑢(0,𝜉)=̂𝑔(𝜉),𝜉𝑛,̂𝑢𝑥(0,𝜉)=0,𝜉𝑛.(1.5) It is easy to obtain the solution of problem (1.5) (if exists) has the form𝑥̂𝑢(𝑥,𝜉)=cosh||𝜉||2𝑘2̂𝑔(𝜉),(1.6) or equivalently, the solution of problem (1.1) has the representation1𝑢(𝑥,𝑦)=(2𝜋)𝑛/2𝑛𝑒𝑖𝜉𝑦𝑥cosh||𝜉||2𝑘2̂𝑔(𝜉)𝑑𝜉.(1.7) Since cosh(𝑥|𝜉|2𝑘2) increases rapidly with exponential order as |𝜉|, the Fourier transform of the exact data 𝑔(𝑦) must decay rapidly. However, in practice, the data at 𝑥=0 is often obtained on the basis of reading of physical instrument which is denoted by 𝑔𝑚. We assume that 𝑔() and 𝑔𝑚() satisfy𝑔()𝑔𝑚()𝐻𝑟𝛿.(1.8) Since 𝑔𝑚() belong to 𝐿2(𝑛)𝐻𝑟(𝑛) for 𝑟0, 𝑟 should not be positive. A small perturbation in the data 𝑔(𝑦) may cause a dramatically large error in the solution 𝑢(𝑥,𝑦) for 0<𝑥1. Hence problem (1.1) is severely ill posed and its numerical simulation is very difficult. It is obvious that the ill-posedness of the problem is caused by the perturbation of high frequencies.

By (1.6) we knoŵ𝑢(1,𝜉)=cosh||𝜉||2𝑘2̂𝑔(𝜉).(1.9) Since the convergence rates can only be given under a priori assumptions on the exact solution [15], we will formulate such an a priori assumption in terms of the exact solution at 𝑥=1 by considering𝑢(1,)𝐻𝑠𝐸.(1.10)

Meyer wavelets are special because, unlike most other wavelets, they have compact support in the frequency domain but not in the time domain (however, they decay very fast). The wavelet methods have been used to solve one-dimensional heat conduction problems [16, 17] and noncharacteristic Cauchy problem for parabolic equation in one-dimensional [18] and multidimensional [19] cases, and so forth. In this paper we propose a similar wavelet method as suggested in [19] to the problem (1.1).

The paper is organized as follows. In Section 2 we describe the Meyer wavelets and discuss the properties that make them useful for solving ill-posed problems. Some error estimates between the exact solution and its approximation as well as the choice of the regularization parameter are given in Section 3. Finally, in Section 4 numerical tests verify the efficiency and accuracy of the proposed method.

2. The Meyer Wavelets

In the present paper let Φ be Meyer’s orthonormal scaling function in 𝑛 dimensions. This function is constructed from the one-dimensional scaling functions in the following way. Let 𝜙(𝑥) and 𝜓(𝑥) be the Meyer scaling and wavelet function in one dimension defined by their Fourier transform in [20] which satisfy4supp𝜙=34𝜋,3𝜋,8supp𝜓=32𝜋,3𝜋23,83𝜋.(2.1) It can be proved (cf. [20]) that the set of functions𝜓𝑗𝑘(𝑥)=2𝑗/2𝜓2𝑗𝑥𝑘,𝑗,𝑘,(2.2) is an orthonormal basis of 𝐿2(). Consequently, the MRA {𝑉𝑗}𝑗 of Meyer is generated by𝑉𝑗={𝜙𝑗𝑘,𝑘},𝜙𝑗𝑘=2𝑗/2𝜙2𝑗𝜙𝑥𝑘,𝑗,𝑘,supp𝑗𝑘=||𝜉||4𝜉;3𝜋2𝑗.(2.3)

For the construction of an 𝑛-dimensional MRA, we take tensor products of the spaces 𝑉𝑗 (see [21, 22]). Then the scaling function Φ is given byΦ(𝑥)=𝑛𝑘=1𝜙𝑥𝑘,𝑥𝑛,(2.4) and any basis function Ψ in 𝑊𝐽 can be written in the formΨ(𝑥)=2𝑛𝐽/2𝜓2𝐽𝑥𝑖𝑘𝑖𝑚𝑖𝜃𝑚2𝐽𝑥𝑚𝑘𝑚,𝑥𝑛,(2.5) where 𝑘𝑛, and for any 𝑚{1,,𝑛}, 𝜃𝑚 stands for 𝜙 or 𝜓. Hence we obtain from (2.1) thatΦ=4supp34𝜋,3𝜋𝑛,(2.6)𝑓(𝜉)=0for𝜉23𝜋2𝐽,𝑓𝑊𝐽,𝐽.(2.7) The orthogonal projection on the space 𝑉𝐽 is defined by𝑃𝐽𝑓=𝑘𝑛𝑓,Φ𝐽,𝑘Φ𝐽,𝑘,(2.8) while 𝑄𝐽𝑓 denotes the orthogonal projection of a function 𝑓 on the wavelet space 𝑊𝐽 with 𝑉𝐽+1=𝑉𝐽𝑊𝐽. (In many contexts one will find more than one detailed space 𝑊𝐽, that is, 𝑉𝐽+1=𝑉𝐽𝑊1,𝐽𝑊2,𝐽. Here, the space 𝑊𝐽 is simply defined as the orthogonal complement of 𝑉𝐽 in 𝑉𝐽+1).

LetΩ𝐽=2𝐽232𝜋,3𝜋𝑛.(2.9) Setting Γ𝐽=𝑛Ω𝐽, together with (2.6), it follows for 𝐽 that𝑃𝐽𝑓(𝜉)=0for𝜉Γ𝐽+1,𝐼𝑃𝐽𝑓̂𝑄(𝜉)=𝐽𝑓(𝜉)for𝜉Ω𝐽+1.(2.10) We introduce the operator 𝑀𝐽 which is defined by the equation𝑀𝐽𝑓=1𝜒𝐽𝑓,𝐽,(2.11) where 𝜒𝐽 denotes the characteristic function of the cube Ω𝐽. From (2.7) it follows that any basis function Ψ in 𝑊𝑗, 𝑗𝐽, satisfiesΨ(𝜉)=0,𝜉Ω𝐽,(2.12) and we obtainΨ=(𝑓,Ψ)=𝑓,1𝜒𝐽Ψ=𝑀𝑓,𝐽𝑓,Ψ.(2.13) And it follows for 𝐽 that𝑄𝐽=𝑄𝐽𝑀𝐽,𝐼𝑃𝐽=𝐼𝑃𝐽𝑀𝐽.(2.14)

3. Wavelet Regularization and Error Estimates

We list the following two lemmas given in [19, 23] which are useful to our proof.

Lemma 3.1 (see [19, 23]). Let {𝑉𝐽}𝐽 be an m-regular MRA, and let 𝑟,𝑠 be such that 𝑚<𝑟<𝑠<𝑚. Then for each function 𝑓𝐻𝑠(𝑛) and 𝐽, the following inequality holds: 𝑓𝑃𝐽𝑓𝐻𝑟𝐶12𝐽(𝑠𝑟)𝑓𝐻𝑟.(3.1)

Lemma 3.2 (see [19]). Let {𝑉𝐽}𝐽 be Meyer’s (tensor-) MRA, and suppose 𝐽, 𝑟. Then for all 𝑓𝑉𝐽, one has 𝜕𝑙𝜕𝑥𝑙𝑖𝑓𝐻𝑟𝐶22(𝐽1)𝑙𝑓𝐻𝑟,𝑖=1,,𝑛,𝑙.(3.2)

Define an operator 𝑇𝑥𝑔(𝑦)𝑢(𝑥,𝑦) by (1.6), that is,𝑇𝑥𝑔=𝑢(𝑥,𝑦),0<𝑥1,(3.3) or equivalently,𝑇𝑥𝑥𝑔(𝜉)=cosh||𝜉||2𝑘2̂𝑔(𝜉),0<𝑥1.(3.4) Then we have

Theorem 3.3. Let {𝑉𝐽}𝐽 be Meyer’s MRA and suppose 𝑟 and 𝐽 which satisfies 2𝐽>𝑘, 0𝑥1. Then for all 𝑓𝑉𝐽, one has 𝑇𝑥𝑓𝐻𝑟𝐶5𝑒𝑥22(𝐽1)𝑘2+1𝑓𝐻𝑟.(3.5)

Proof. For 𝑓𝑉𝐽, by definition (1.4) and formula (3.4), from Lemma 3.2, we have 𝑇𝑥𝑓𝐻𝑟=𝑛||||cosh(𝑥||𝜉||2𝑘2)𝑓||||2||𝜉||1+2𝑟𝑑𝜉1/2|𝜉|>𝑘||||||||𝑥cosh||𝜉||2𝑘2𝑥||𝜉||𝑥||𝜉||𝑓||||||||coshcosh2||𝜉||1+2𝑟𝑑𝜉1/2+|𝜉|𝑘||||𝑥cos𝑘2||𝜉||2𝑓||||2||𝜉||1+2𝑟𝑑𝜉1/2sup𝜉Ω𝐽+1||||||||𝑥cosh||𝜉||2𝑘2𝑥||𝜉||||||||||cosh|𝜉|>𝑘||𝑥||𝜉||𝑓||cosh2||𝜉||1+2𝑟𝑑𝜉1/2+𝑓𝐻𝑟2sup𝜉Ω𝐽+1𝑒𝑥(|𝜉|2𝑘2|𝜉|)|𝜉|>𝑘|||||𝑙=0𝑥2𝑙(2𝑙)!|𝜉|2𝑙𝑓|||||2||𝜉||1+2𝑟𝑑𝜉1/2+𝑓𝐻𝑟2𝐶3𝑒𝑥(22(𝐽1)𝑘22𝐽1)𝑙=0𝑥2𝑙(2𝑙)!(Δ𝑦)𝑙𝑓𝐻𝑟+𝑓𝐻𝑟𝐶4𝑒𝑥(22(𝐽1)𝑘22𝐽1)𝑙=0𝑥2𝑙(2𝑙)!𝑛22(𝐽1)𝑙+1𝑓𝐻𝑟𝐶5𝑒𝑥(22(𝐽1)𝑘22𝐽1)cosh𝑥2𝐽1+1𝑓𝐻𝑟𝐶5𝑒𝑥22(𝐽1)𝑘2+1𝑓𝐻𝑟.(3.6)

Since the Cauchy data are given inexactly by 𝑔𝑚, we need a stable algorithm to approximate the solution of (1.1). Our method is as follows. Consider the operator𝑇𝑥,𝐽=𝑃𝐽𝑇𝑥𝑃𝐽,(3.7) and show that it approximates 𝑇𝑥 in a stable way for an appropriate choice for 𝐽 depending on 𝛿 and 𝐸. By the triangle inequality we know𝑇𝑥𝑔𝑇𝑥,𝐽𝑔𝑚𝐻𝑟(𝑇𝑥𝑇𝑥,𝐽)𝑔𝐻𝑟+𝑇𝑥,𝐽𝑔𝑔𝑚𝐻𝑟.(3.8)

From (1.8) and Theorem 3.3, the second term on the right-hand side of (3.8) satisfies𝑇𝑥,𝐽𝑔𝑔𝑚𝐻𝑟=𝑃𝐽𝑇𝑥𝑃𝐽𝑔𝑔𝑚𝐻𝑟𝑇𝑥𝑃𝐽(𝑔𝑔𝑚)𝐻𝑟𝐶5𝑒𝑥22(𝐽1)𝑘2+1𝛿.(3.9)

For the first one we have𝑇𝑥𝑇𝑥,𝐽𝑔𝐻𝑟𝐼𝑃𝐽𝑇𝑥𝑔𝐻𝑟+𝑃𝐽𝑇𝑥𝐼𝑃𝐽𝑔𝐻𝑟.(3.10)

By Lemma 3.1, (1.9), (1.10), (2.14), and (3.4), we get𝐼𝑃𝐽𝑇𝑥𝑔𝐻𝑟=𝐼𝑃𝐽𝑀𝐽𝑇𝑥𝑔𝐻𝑟𝐶12𝐽(𝑠𝑟)𝑀𝐽𝑇𝑥𝑔𝐻𝑠=𝐶12𝐽(𝑠𝑟)Γ𝐽||||||||𝑥cosh||𝜉||2𝑘2cosh||𝜉||2𝑘2||||||||𝑢(1,)2||𝜉||1+2𝑠1/2𝐶12𝐽(𝑠𝑟)sup𝜉Γ𝐽||||||||𝑥cosh||𝜉||2𝑘2cosh||𝜉||2𝑘2||||||||𝑢(1,)𝐻𝑠2𝐶132𝜋2𝐽2𝑘2(𝑠𝑟)/2𝑒(1𝑥)((2/3)𝜋2𝐽)2𝑘2𝐸2𝐶122𝐽𝑘2(𝑠𝑟)/2𝑒(1𝑥)22𝐽𝑘2𝐸.(3.11) On the other hand, due to (2.10), we know𝑃𝐽𝑇𝑥𝐼𝑃𝐽𝑔𝐻𝑟𝑇𝑥𝐼𝑃𝐽𝑔𝐻𝑟Ω𝐽+1||||𝑥cosh||𝜉||2𝑘2𝑄𝐽𝑔̂||||(𝜉)2||𝜉||1+2𝑟𝑑𝜉1/2+Γ𝐽+1||||𝑥cosh||𝜉||2𝑘2||||̂𝑔(𝜉)2||𝜉||1+2𝑟𝑑𝜉1/2=𝐼1+𝐼2.(3.12) We estimate the two parts at the right-hand side of (3.12) separately. For 𝐼2 we have𝐼2=Γ𝐽+1||||𝑥cosh||𝜉||2𝑘2||||̂𝑔(𝜉)2||𝜉||1+2𝑟𝑑𝜉1/2=Γ𝐽+1||||||||𝑥cosh||𝜉||2𝑘2cosh||𝜉||2𝑘2||||||||𝑢(1,)2||𝜉||1+2𝑟𝑑𝜉1/2sup𝜉Γ𝐽+1||||||||𝑥cosh||𝜉||2𝑘2cosh||𝜉||2𝑘21||𝜉||1+2(𝑠𝑟)/2||||||||×Γ𝐽+1||||||𝜉||𝑢(1,)1+2𝑠/2||||2𝑑𝜉1/2423𝜋2𝐽(𝑠𝑟)𝑒(1𝑥)𝑛((4/3)𝜋2𝐽)2𝑘2𝑢(1,)𝐻𝑠2(22𝐽𝑘2)(𝑠𝑟)/2𝑒(1𝑥)22𝐽𝑘2𝐸.(3.13) Now we turn to 𝐼1. There holds𝐼1=Ω𝐽+1||||𝑥cosh||𝜉||2𝑘2𝑄𝐽𝑔̂||||(𝜉)2||𝜉||1+2𝑟𝑑𝜉1/2𝑇𝑥𝑄𝐽𝑔𝐻𝑟𝐶5𝑒𝑥22𝐽𝑘2𝑄+1𝐽𝑔𝐻𝑟,(3.14) since 𝑄𝐽𝑔𝑉𝐽+1. Furthermore, from (2.14), it follows that𝑄𝐽𝑔𝐻𝑟=𝑄𝐽𝑀𝐽𝑔𝐻𝑟𝑀𝐽𝑔𝐻𝑟=Γ𝐽||||̂𝑔(𝜉)2||𝜉||1+2𝑟𝑑𝜉1/2=Γ𝐽||||||||̂𝑢(1,𝜉)cosh||𝜉||2𝑘2||||||||2||𝜉||1+2𝑟𝑑𝜉1/222𝐽(𝑠𝑟)𝑒𝑛((2/3)𝜋2𝐽)2𝑘2𝑢(1,)𝐻𝑠2(22𝐽𝑘2)(𝑠𝑟)/2𝑒22𝐽𝑘2𝐸.(3.15) Therefore,𝑃𝐽𝑇𝑥(𝐼𝑃𝐽)𝑔𝐻𝑟21+𝐶522𝐽𝑘2(𝑠𝑟)/2𝑒(1𝑥)22𝐽𝑘2𝐸2+22𝐽𝑘2(𝑠𝑟)/2𝑒22𝐽𝑘2𝐸.(3.16) Combining (3.11) and (3.16) with (3.10), we have(𝑇𝑥𝑇𝑥,𝐽)𝑔𝐻𝑟𝐶21+1+𝐶522𝐽𝑘2(𝑠𝑟)/2𝑒(1𝑥)22𝐽𝑘2𝐸2+22𝐽𝑘2(𝑠𝑟)/2𝑒22𝐽𝑘2𝐸.(3.17) Then from (3.9) and (3.17) we finally arrive at𝑇𝑥𝑔𝑇𝑥,𝐽𝑔𝑚𝐻𝑟𝐶5𝑒𝑥22(𝐽1)𝑘22+1𝛿+22𝐽𝑘2(𝑠𝑟)/2𝑒22𝐽𝑘2𝐸𝐶+21+1+𝐶522𝐽𝑘2(𝑠𝑟)/2𝑒(1𝑥)22𝐽𝑘2𝐸.(3.18)

In order to show some stability estimates of the Hölder type for our method using (3.18), we use the following lemma which appeared in [24] for choosing a proper regularization parameter 𝐽.

Lemma 3.4. Let the function 𝑓(𝜆)[0,𝑎] be given by 𝑓(𝜆)=𝜆𝑏1𝑑ln𝜆𝑐(3.19) with a constant 𝑐 and positive constants 𝑎<1, 𝑏, and 𝑑. Then for the inverse function 𝑓1(𝜆), one has 𝑓1(𝜆)=𝜆1/𝑏𝑑𝑏1ln𝜆𝑐/𝑏(1+𝑜(1))for𝜆0.(3.20)

Based on this lemma, we can choose the regularization parameter 𝐽 by minimizing the right-hand side of (3.18).

Denote𝑒22𝐽𝑘2=𝜆(0,1),(3.21) and let 𝐶=𝐶5/2(1+𝐶1+𝐶5) and𝐶𝜆𝑥𝛿=𝜆1𝑥1ln𝜆(𝑠𝑟)𝐸,(3.22) that is,𝐶𝛿𝐸1=𝜆ln𝜆(𝑠𝑟).(3.23) Then by Lemma 3.4 we obtain that𝜆=𝐶𝛿𝐸1ln𝐶𝛿/𝐸𝑠𝑟(1+𝑜(1))for𝐶𝛿𝐸=0𝐶𝛿𝐸𝐸ln𝐶𝛿𝑠𝑟(1+𝑜(1))for𝛿0.(3.24) Taking the principal part of 𝜆, we get𝐽=12log2ln2𝐸𝐸𝐶𝛿ln𝐶𝛿(𝑠𝑟)+𝑘2,(3.25) due to (3.21). Now, summarizing above inference process, we obtain the main result of the present paper.

Theorem 3.5. For 𝑠𝑟, suppose that conditions (1.8) and (1.10) hold. If one takes 𝐽=𝐽,(3.26) where 𝐽 was defined in (3.25), [𝑎] with square bracket denotes the largest integer less than or equal to 𝑎𝑅. Then there holds the following stability estimate: 𝑇𝑥𝑔𝑇𝑥,𝐽𝑔𝑚2𝐶1+1+𝐶5𝐸𝑥𝐶5𝛿1𝑥𝐸ln𝐶𝛿(𝑠𝑟)𝑥×1+ln𝐸/𝐶𝛿ln𝐸/𝐶𝛿+ln(ln𝐸/𝐶𝛿)(𝑠𝑟)𝑠𝑟+1+2𝐶ln𝐸/𝐶𝛿ln𝐸/𝐶𝛿+ln(ln𝐸/𝐶𝛿)(𝑠𝑟)𝑠𝑟𝛿=2𝐶1+1+𝐶5𝐸𝑥𝐶5𝛿1𝑥𝐸ln𝐶𝛿(𝑠𝑟)𝑥(1+𝑜(1)),(3.27) for 𝛿0.

Remark 3.6. In general, the a priori bound 𝐸 and the coefficients 𝐶1-𝐶5 and 𝐶 are not exactly known in practice. In this case, with 𝐽=12log2ln21𝛿1ln𝛿(𝑠𝑟)+𝑘2,(3.28) it holds that 𝑇𝑥𝑔𝑇𝑥,𝐽𝑔𝑚𝛿1𝑥1ln𝛿(𝑠𝑟)𝑥𝐶5𝐶+21+1+𝐶5𝐸ln1/𝛿ln1/𝛿+ln(ln1/𝛿)(𝑠𝑟)𝑠𝑟+𝐶5+2𝐸ln1/𝛿ln1/𝛿+ln(ln1/𝛿)(𝑠𝑟)𝑠𝑟𝛿=𝛿1𝑥1ln𝛿(𝑠𝑟)𝑥(1+𝑜(1)),(3.29) for 𝛿0.

Remark 3.7. The proposed wavelet method can also be used to solve the following Cauchy problem for the modified Helmholtz equation (i.e., the Yukawa equation [25]) Δ𝑣(𝑥,𝑦)+𝑘2𝑣(𝑥,𝑦)=0,𝑥(0,1),𝑦𝑛,𝑛1,𝑣(0,𝑦)=𝑔(𝑦),𝑦𝑛,𝑣𝑥(0,𝑦)=0,𝑦𝑛,(3.30) where Δ=𝜕2/𝜕𝑥2+𝑛𝑖=1𝜕2/Δ𝑦2𝑖 is the same as in (1.1).
It is easy to know that the exact solution of problem (3.30) is 1𝑣(𝑥,𝑦)=(2𝜋)𝑛/2𝑛𝑒𝑖𝜉𝑦𝑥cosh||𝜉||2+𝑘2̂𝑔(𝜉)𝑑𝜉.(3.31) Define an operator 𝑇𝑥𝑔(𝑦)𝑣(𝑥,𝑦) such that 𝑇𝑥𝑥𝑔(𝜉)=cosh||𝜉||2+𝑘2̂𝑔(𝜉),0<𝑥1,(3.32) and the approximate solution is 𝑣𝛿𝐽=𝑇𝑥,𝐽𝑔𝑚,(3.33) where 𝑔 and 𝑔𝑚 satisfy (1.8), and 𝑇𝑥,𝐽=𝑃𝐽𝑇𝑥𝑃𝐽. If we select the regularization parameter 𝐽=12log2ln21𝛿1ln𝛿(𝑠𝑟)𝑘2,(3.34) then there holds 𝑣(𝑥,)𝑣𝛿𝐽(𝑥,)𝛿1𝑥1ln𝛿(𝑠𝑟)𝑥×𝐶5𝐶+21+1+𝐶5𝐸ln1/𝛿ln1/𝛿+ln(ln1/𝛿)(𝑠𝑟)𝑠𝑟+𝐶5+2𝐸ln1/𝛿ln1/𝛿+ln(ln1/𝛿)(𝑠𝑟)𝑠𝑟𝛿=𝛿1𝑥1ln𝛿(𝑠𝑟)𝑥(1+(𝑜(1))),(3.35) for 𝛿0.

4. Numerical Aspect

4.1. Numerical Implementation

We want to discuss some numerical aspects of the proposed method in this section.

We consider the case when 𝑛=2. Supposing that the sequence {𝑔(𝑦1,𝑖,𝑦2,𝑗)}𝑁𝑖,𝑗=1 represents samples from the function 𝑔(𝑦1,𝑦2) on an equidistant grid in the square [𝑎,𝑏]2, and 𝑁 is even, then we add a random uniformly distributed perturbation to each data and obtain the perturbation data𝑔𝑚=𝑔+𝜇randn(size(𝑔)).(4.1) Then the total noise 𝛿 can be measured in the sense of root mean square error according to𝑔𝛿=𝑚𝑔𝑙2=1𝑁2𝑁𝑁𝑖=1𝑗=1𝑔𝑚𝑦1,𝑖,𝑦2,𝑗𝑦𝑔1,𝑖,𝑦2,𝑗2,(4.2) where randn() is a normally distributed random variable with zero mean and unit standard deviation and 𝜖 dictates the level of noise. randn(size(𝑔)) returns an array of random entries that is the same size as 𝑔.

For the function 𝑔𝑚(𝑦1,𝑦2), we have𝑢𝛿𝐽(𝑥,𝑦)=𝑇𝑥,𝐽𝑔𝑚=𝑃𝐽𝑇𝑥𝑃𝐽𝑔𝑚.(4.3) Hence, by using it with 𝐽 being given in (3.28), we can obtain the approximate solution.

We will use DMT as a short form of the “discrete Meyer (wavelet) transform." Algorithms for discretely implementing the Meyer wavelet transform are described in [21]. These algorithms are based on the fast Fourier transform (FFT), and computing the DMT of a vector in requires 𝒪(𝑁log22𝑁) operations.

4.2. Numerical Tests

In this section some numerical tests are presented to demonstrate the usefulness of the approach. The tests were performed using Matlab and the wavelet package WaveLab 850, which was downloaded from http://www-stat.stanford.edu/~wavelab/. Throughout this section, we set 𝜇=103, 𝑎=5, 𝑏=5, and 𝑁=26.

Example 4.1. Take 𝑛=2 and 𝑔(𝑦)=𝑒𝑦2𝒮(2), where 𝑦=(𝑦1,𝑦2) and 𝒮(2) denotes the Schwartz function space.
Since ̂𝑔(𝜉)𝒮(2), 𝜉=(𝜉1,𝜉2) decays rapidly, and the formula (1.7) can be used to calculate 𝑢(𝑥,𝑦) with exact data directly, that is, 1𝑢(𝑥,𝑦)=2𝜋2𝑒𝑖(𝜉1𝑦1+𝜉2𝑦2)𝑥cosh𝜉21+𝜉22𝑘2𝜉̂𝑔1,𝜉2𝑑𝜉1𝑑𝜉2.(4.4)

In Figure 1 we give the exact solution at 𝑥=1, that is, 𝑢(1,𝑦1,𝑦2), and the reconstructed solution 𝑢𝛿(1,𝑦1,𝑦2) from the noisy data 𝑔𝑚(𝑦1,𝑦2) without regularization. We see that 𝑢𝛿 does not approximate the solution and some regularization procedure is necessary.

Letting 𝑘=1, the regularized solutions and the corresponding errors 𝑢𝑢𝛿𝐽 defined by the regularization parameter 𝐽=3,4,5 are illustrated in Figure 2. We can see that in 𝑉3 the approximation is very poor since the frequencies are cut off excessively by the projection 𝑃3. If 𝐽 is taken to be too large, the noise in the function 𝑔𝑚 is not damped enough by 𝑃𝐽, and thus the high frequencies of ̂𝑔𝑚 are so extremely magnified that they destroy the approximated solution. The approximation parameter 𝐽=4 seems to be the optimal choice for this example.

In Figure 3 we display the exact solution, its approximation, and corresponding errors for 𝑘=5 and 100, respectively. We see that the proposed method is useful for different wave number 𝑘.

Figure 4 shows that the proposed method for the Cauchy problem for the modified Helmholtz equation is also effective.

Acknowledgments

The authors would like to thank the WaveLab Team at the Stanford University for the help of their wavelet package Wavelab850. The work described in this paper was supported by the Fundamental Research Funds for the Central Universities of China (Project no. ZYGX2009J099) and the National Natural Science Foundation of China (Project no. 11171136).

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Copyright © 2012 Fang-Fang Dou and Chu-Li Fu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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