Open Access

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

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 https://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 https://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]:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ1_HTML.gif
(1.1)
However, the boundary condition https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq4_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq5_HTML.gif is a function of https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq7_HTML.gif from the Cauchy data pairs https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq8_HTML.gif located at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq9_HTML.gif . Of course, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq10_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq12_HTML.gif , for which the measurement errors https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq13_HTML.gif and https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq15_HTML.gif denotes the https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq17_HTML.gif given on the line https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq19_HTML.gif be the Fourier transform of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq20_HTML.gif :
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq21_HTML.gif :
https://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
https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ6_HTML.gif
(2.4)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq24_HTML.gif . The solution https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq26_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq27_HTML.gif . The difference between the exact solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq28_HTML.gif and the regularized solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq29_HTML.gif can be divided into two parts:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ7_HTML.gif
(2.5)
where
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ9_HTML.gif
(2.7)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq30_HTML.gif , we can rewrite (2.7) as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ10_HTML.gif
(2.8)
where the matrix https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq32_HTML.gif instead of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq33_HTML.gif in the definition of https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq36_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq37_HTML.gif . Thus we can factorize https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq38_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ11_HTML.gif
(2.9)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq39_HTML.gif is a unitary matrix, https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ12_HTML.gif
(2.10)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq41_HTML.gif is unitary, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq42_HTML.gif and therefore
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ13_HTML.gif
(2.11)
where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq44_HTML.gif and the subordinate matrix norm, remembering https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq45_HTML.gif in (2.11), we obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq46_HTML.gif and we can integrate over https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq48_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq49_HTML.gif in the https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq51_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq53_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq55_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq56_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ15_HTML.gif
(2.13)

Proof.

The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq57_HTML.gif satisfies the differential equation; thus https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ16_HTML.gif
(2.14)
and thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq59_HTML.gif satisfies inequality(2.12). We have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ17_HTML.gif
(2.15)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq60_HTML.gif , then
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ19_HTML.gif
(2.17)
If we insert https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq61_HTML.gif and integrate over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq62_HTML.gif , then we obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq63_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq66_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ21_HTML.gif
(2.19)

Proof.

The function https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ23_HTML.gif
(2.21)
and thus, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq68_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq69_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ25_HTML.gif
(2.23)
Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq71_HTML.gif , we can extend the integrals. Inserting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq72_HTML.gif we get
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq74_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq76_HTML.gif ; then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ27_HTML.gif
(2.25)

Proof.

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq78_HTML.gif . Then by Lemma 2.1,
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ29_HTML.gif
(2.27)
By the triangle inequality,
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq80_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq81_HTML.gif we, assume that the Helmholtz equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq82_HTML.gif is valid in a large interval https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq84_HTML.gif and at https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq86_HTML.gif and a regularized solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq87_HTML.gif with "cutoff'' level https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq89_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq90_HTML.gif for the case of exact data https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq93_HTML.gif . Then the difference between https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq94_HTML.gif and a regularized solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq95_HTML.gif can be estimated:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ31_HTML.gif
(2.29)
where the constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq96_HTML.gif is defined by
https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ33_HTML.gif
(2.31)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq97_HTML.gif and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ34_HTML.gif
(2.32)
Solving for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq99_HTML.gif and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ35_HTML.gif
(2.33)
where
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ37_HTML.gif
(2.35)
and that
https://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
https://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
https://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:
https://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
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq103_HTML.gif , along the line https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq105_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq106_HTML.gif , and compute https://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 https://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 https://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 https://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:
https://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:
https://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 https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq112_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq113_HTML.gif and if we choose https://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),
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq115_HTML.gif and used the bound https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq116_HTML.gif . By expression (2.41),
https://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:
https://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, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq119_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq121_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq122_HTML.gif . Then we have the estimates:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq123_HTML.gif and https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ50_HTML.gif
(3.2)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Equ51_HTML.gif
(3.3)

The matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq125_HTML.gif approximates the second-order derivative https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq128_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq129_HTML.gif from the Cauchy data pairs https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq130_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_IEq132_HTML.gif and different locations https://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 https://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 https://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 https://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.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F212056/MediaObjects/13661_2009_Article_905_Fig1_HTML.jpg

Figure 1

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

Figure 2

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

Figure 3

https://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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