A Spectral Regularization Method for a Cauchy Problem of the Modified Helmholtz Equation

  • Ailin Qian1Email author,

    Affiliated with

    • Jianfeng Mao1 and

      Affiliated with

      • Lianghua Liu1

        Affiliated with

        Boundary Value Problems20102010:212056

        DOI: 10.1155/2010/212056

        Received: 15 December 2009

        Accepted: 9 May 2010

        Published: 13 June 2010

        Abstract

        We consider the Cauchy problem for a modified Helmholtz equation, where the Cauchy data is given at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq1_HTML.gif and the solution is sought in the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq2_HTML.gif . A spectral method together with choice of regularization parameter is presented and error estimate is obtained. Combining the method of lines, we formulate regularized solutions which are stably convergent to the exact solutions.

        1. Introduction

        The Cauchy problem for Helmholtz equation arises from inverse scattering problems. Specific backgrounds can be seen in the existing literature; we can refer to [16] and so forth. A number of numerical methods for stabilizing this problem are developed. Several boundary element methods combined with iterative, conjugate gradient, Tikhonov regularization, and singular value decomposition methods are compared in [6]. Cauchy problem for elliptic equations is well known to be severely ill-posed [7]; that is, the solution does not depend continuously on the boundary data, and small errors in the boundary data can amplify the numerical solution infinitely; hence it is impossible to solve Cauchy problem of Helmholtz equation by using classical numerical methods and it requires special techniques, for example, regularization methods. Although theoretical concepts and computational implementation related to the Cauchy problem of Helmholtz equation have been discussed by many authors [811], there are many open problems deserved to be solved. For example, many authors have considered the following Cauchy problem of the Helmholtz equation [811]:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ1_HTML.gif
        (1.1)
        However, the boundary condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq3_HTML.gif is very strict. If the boundary condition is replaced by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq4_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq5_HTML.gif is a function of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq6_HTML.gif , then their methods cannot be applied easily. Therefore, in this paper, we consider the following Cauchy problem for the Helmholtz equation:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ2_HTML.gif
        (1.2)

        We want to seek the solution in the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq7_HTML.gif from the Cauchy data pairs http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq8_HTML.gif located at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq9_HTML.gif . Of course, since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq11_HTML.gif are assumed to be measured, there must be measurement errors, and we would actually have noisy data function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq12_HTML.gif , for which the measurement errors http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq14_HTML.gif are small. Here and in the following sections, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq15_HTML.gif denotes the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq16_HTML.gif norm. Thus (1.2) is a noncharacteristic Cauchy problem with appropriate Cauchy data http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq17_HTML.gif given on the line http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq18_HTML.gif .

        However, the ill-posedness is caused by high frequency. By introducing a "cutoff'' frequency we can obtain a well-posed problem. This method has been used for solving inverse heat conduction problem [12], and sideways heat equation [13], sideways parabolic equation [14]. An error estimate for the proposed method can be found in Section 2. The implementation of the numerical method is explained.

        2. Regularization and Error Estimate

        First, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq19_HTML.gif be the Fourier transform of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq20_HTML.gif :
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ3_HTML.gif
        (2.1)
        Taking Fourier transformation for (1.2), we have a family of problems parameterized by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq21_HTML.gif :
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ4_HTML.gif
        (2.2)
        The solution can easily be verified to be
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ5_HTML.gif
        (2.3)
        Following the idea of Fourier method, we consider (2.2) only for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq22_HTML.gif by cutting off high frequency and define a regularized solution:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ6_HTML.gif
        (2.4)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq23_HTML.gif is the characteristic function of the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq24_HTML.gif . The solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq25_HTML.gif can be found by using the inverse Fourier transform. Define the regularized solution with measured data http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq26_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq27_HTML.gif . The difference between the exact solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq28_HTML.gif and the regularized solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq29_HTML.gif can be divided into two parts:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ7_HTML.gif
        (2.5)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ8_HTML.gif
        (2.6)
        We rewrite (2.2) as a system of ordinary differential equations:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ9_HTML.gif
        (2.7)
        Letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq30_HTML.gif , we can rewrite (2.7) as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ10_HTML.gif
        (2.8)
        where the matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq31_HTML.gif is the one in (2.7). The reason for using http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq32_HTML.gif instead of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq33_HTML.gif in the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq34_HTML.gif is that with this choice, the matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq35_HTML.gif is normal and hence diagonalized by a unitary matrix. The eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq36_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq37_HTML.gif . Thus we can factorize http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq38_HTML.gif as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ11_HTML.gif
        (2.9)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq39_HTML.gif is a unitary matrix, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq40_HTML.gif it follows that the solution of the system (2.8) can be written as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ12_HTML.gif
        (2.10)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq41_HTML.gif is unitary, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq42_HTML.gif and therefore
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ13_HTML.gif
        (2.11)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq43_HTML.gif denotes both the Euclidean norm in the complex vector space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq44_HTML.gif and the subordinate matrix norm, remembering http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq45_HTML.gif in (2.11), we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ14_HTML.gif
        (2.12)

        This inequality is valid for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq46_HTML.gif and we can integrate over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq47_HTML.gif and use the Parseval theorem to obtain estimates for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq48_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq49_HTML.gif in the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq50_HTML.gif -norm. First we will prove a bound on the difference between any two regularized solutions (2.4). We have errors in the measured http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq51_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq52_HTML.gif . These two cases are treated separately.

        Lemma 2.1.

        Assume that one has two regularized solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq53_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq54_HTML.gif defined by (2.4), with the Cauchy data http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq56_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ15_HTML.gif
        (2.13)

        Proof.

        The function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq57_HTML.gif satisfies the differential equation; thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq58_HTML.gif solves (2.2), with initial data given by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ16_HTML.gif
        (2.14)
        and thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq59_HTML.gif satisfies inequality(2.12). We have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ17_HTML.gif
        (2.15)
        If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq60_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ18_HTML.gif
        (2.16)
        holds. By inserting (2.15) into (2.16) we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ19_HTML.gif
        (2.17)
        If we insert http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq61_HTML.gif and integrate over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq62_HTML.gif , then we obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ20_HTML.gif
        (2.18)

        Thus (2.13) holds.

        Lemma 2.2.

        Assume that one has two regularized solutions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq63_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq64_HTML.gif defined by (2.4), with the Cauchy data http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq66_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ21_HTML.gif
        (2.19)

        Proof.

        The function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq67_HTML.gif satisfies the differential equation, with initial data given by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ22_HTML.gif
        (2.20)
        and thus inequality (2.12) holds. It follows that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ23_HTML.gif
        (2.21)
        and thus, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq68_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ24_HTML.gif
        (2.22)
        By integrating over the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq69_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ25_HTML.gif
        (2.23)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq70_HTML.gif is equal to zero outside the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq71_HTML.gif , we can extend the integrals. Inserting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq72_HTML.gif we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ26_HTML.gif
        (2.24)

        This is precisely (2.19).

        Next we prove that, for the regularized problem, we have stable dependence of the data. By using the two previous lemmas, we get the following.

        Lemma 2.3 (stability).

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq73_HTML.gif is the regularized solution (2.4), with exact data http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq75_HTML.gif is the regularized solution with noisy data http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq76_HTML.gif ; then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ27_HTML.gif
        (2.25)

        Proof.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq77_HTML.gif be a regularized solution defined by (2.4), with data http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq78_HTML.gif . Then by Lemma 2.1,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ28_HTML.gif
        (2.26)
        Using Lemma 2.2, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ29_HTML.gif
        (2.27)
        By the triangle inequality,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ30_HTML.gif
        (2.28)

        This completes the proof.

        By Lemma 2.3 the regularized solution depends continuously on the data. Next we derive a bound on the truncation error when we neglect frequencies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq79_HTML.gif in (2.3). So far we have not used any information about http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq80_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq81_HTML.gif we, assume that the Helmholtz equation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq82_HTML.gif is valid in a large interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq83_HTML.gif . By imposing a priori bounds on the solution at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq84_HTML.gif and at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq85_HTML.gif , we obtain an estimate of the difference between the exact solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq86_HTML.gif and a regularized solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq87_HTML.gif with "cutoff'' level http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq88_HTML.gif . This is a convergence result in the sense that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq89_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq90_HTML.gif for the case of exact data http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq91_HTML.gif . The following estimate holds.

        Lemma 2.4 (convergence).

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq92_HTML.gif is the solution of the problem (2.2), and that the Helmholtz equation is valid for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq93_HTML.gif . Then the difference between http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq94_HTML.gif and a regularized solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq95_HTML.gif can be estimated:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ31_HTML.gif
        (2.29)
        where the constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq96_HTML.gif is defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ32_HTML.gif
        (2.30)

        Proof.

        The solution of (2.2) can be written in the floowing form:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ33_HTML.gif
        (2.31)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq97_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq98_HTML.gif can be determined from the boundary conditions:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ34_HTML.gif
        (2.32)
        Solving for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq99_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq100_HTML.gif we find that the solution can be written:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ35_HTML.gif
        (2.33)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ36_HTML.gif
        (2.34)
        We make the observation that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ37_HTML.gif
        (2.35)
        and that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ38_HTML.gif
        (2.36)
        Using the expression (2.33) and the triangle inequality, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ39_HTML.gif
        (2.37)
        The first term on the right-hand side satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ40_HTML.gif
        (2.38)
        Similarly, the second term can be estimated:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ41_HTML.gif
        (2.39)
        By combining these two expressions, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ42_HTML.gif
        (2.40)

        Thus the proof is complete.

        Remark 2.5.

        The constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq101_HTML.gif is well defined, but its value has to be estimated. From a numerical computation we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq102_HTML.gif .

        Remark 2.6.

        When solving (2.2) numerically we need Cauchy data http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq103_HTML.gif , along the line http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq104_HTML.gif . The most natural way to obtain this is to use two thermocouples, located at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq105_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq106_HTML.gif , and compute http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq107_HTML.gif by solving a well-posed problem for the Helmholtz equation in the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq108_HTML.gif . Hence it is natural to assume knowledge about http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq109_HTML.gif at a second point.

        Let us summarize what we have so far. The constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq110_HTML.gif in Lemma 2.4 is unchanged. The propagated data error is estimated using Lemma 2.3:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ43_HTML.gif
        (2.41)
        and the truncation error is estimated using Lemma 2.4:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ44_HTML.gif
        (2.42)

        These two results can be combined into an error estimate for the spectral method. This is demonstrated in two examples.

        Example 2.7.

        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq111_HTML.gif , and that we have an estimate of the noise level, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq112_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq113_HTML.gif and if we choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq114_HTML.gif , with this choice, then by expression (2.42),
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ45_HTML.gif
        (2.43)
        where we have assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq115_HTML.gif and used the bound http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq116_HTML.gif . By expression (2.41),
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ46_HTML.gif
        (2.44)
        Thus we obtain an error estimate:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ47_HTML.gif
        (2.45)

        Note that, under these assumptions, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq117_HTML.gif can be used as a rule for selecting the regularization parameter.

        Example 2.8.

        Suppose that the Helmholtz equation is valid in the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq118_HTML.gif and that we have a priori bounds http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq119_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq120_HTML.gif . Furthermore, we assume that the measured data satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq121_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq122_HTML.gif . Then we have the estimates:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ48_HTML.gif
        (2.46)

        By balancing these two components http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq123_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq124_HTML.gif , we can find a suitable value for the regularization parameter.

        3. Numerical Implementation

        In this section, we use the method of Lines. Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ49_HTML.gif
        (3.1)
        and then we can obtain
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ50_HTML.gif
        (3.2)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ51_HTML.gif
        (3.3)

        The matrix http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq125_HTML.gif approximates the second-order derivative http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq126_HTML.gif by spectral cutoff method (see [2]). This ordinary differential equation system can easily be solved by various numerical methods.

        Example 3.1.

        It is easy to verify that the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq127_HTML.gif is the exact solution of (1.2). Now we need to seek the solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq128_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq129_HTML.gif from the Cauchy data pairs http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq130_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq131_HTML.gif .

        In our numerical experiment, we give the comparison between the exact solution of problem (2.2) and its approximation by spectral cutoff method for different noise levels http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq132_HTML.gif and different locations http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq133_HTML.gif . Please see Figures 1, 2, 3, and 4. In the experiment, the regularization parameter is chosen according to the Remark of Remark 2.5, where we take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq134_HTML.gif . From the above results, it is easy to see that the numerical effect of the spectral method works well. Moreover, we can also see that the lower the noise level http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq135_HTML.gif is, the better the approximate effect is; the closer to the boundary http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq136_HTML.gif the location is, the better the approximate effect is. These accord with the theory in Section 2.
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Fig1_HTML.jpg

        Figure 1

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Fig2_HTML.jpg

        Figure 2

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Fig3_HTML.jpg

        Figure 3

        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Fig4_HTML.jpg

        Figure 4

        Declarations

        Acknowledgments

        The author wants to express his thanks to the referee for many valuable comments. This work is supported by the Educational Commission of Hubei Province of China (Q20102804, T201009).

        Authors’ Affiliations

        (1)
        School of Mathematics and Statistics, Xianning College

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        © Ailin Qian et al. 2010

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