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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.

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 [

In this paper we consider the Cauchy problem for the Helmholtz equation in a “strip”

Let

For

We assume there exists a unique solution

By (

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 [

The paper is organized as follows. In Section

In the present paper let

For the construction of an

Let

We list the following two lemmas given in [

Let

Let

Define an operator

Let

For

Since the Cauchy data are given inexactly by

From (

For the first one we have

By Lemma

In order to show some stability estimates of the Hölder type for our method using (

Let the function

Based on this lemma, we can choose the regularization parameter

Denote

For

In general, the a priori bound

The proposed wavelet method can also be used to solve the following Cauchy problem for the modified Helmholtz equation (i.e., the Yukawa equation [

It is easy to know that the exact solution of problem (

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

We consider the case when

For the function

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 [

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

Since

In Figure

(a) Exact solution

Letting

(a) The regularized solution at

In Figure

(a) and (b) correspond to

Figure

(a) Exact solution

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).