A Spectral Regularization Method for a Cauchy Problem of the Modified Helmholtz Equation
© Ailin Qian et al. 2010
Received: 15 December 2009
Accepted: 9 May 2010
Published: 13 June 2010
We consider the Cauchy problem for a modified Helmholtz equation, where the Cauchy data is given at and the solution is sought in the interval . 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.
We want to seek the solution in the interval from the Cauchy data pairs located at . Of course, since and are assumed to be measured, there must be measurement errors, and we would actually have noisy data function , for which the measurement errors and are small. Here and in the following sections, denotes the norm. Thus (1.2) is a noncharacteristic Cauchy problem with appropriate Cauchy data given on the line .
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 , and sideways heat equation , sideways parabolic equation . 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
This inequality is valid for all and we can integrate over and use the Parseval theorem to obtain estimates for and in the -norm. First we will prove a bound on the difference between any two regularized solutions (2.4). We have errors in the measured and . These two cases are treated separately.
Thus (2.13) holds.
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).
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 in (2.3). So far we have not used any information about for we, assume that the Helmholtz equation is valid in a large interval . By imposing a priori bounds on the solution at and at , we obtain an estimate of the difference between the exact solution and a regularized solution with "cutoff'' level . This is a convergence result in the sense that as for the case of exact data . The following estimate holds.
Lemma 2.4 (convergence).
Thus the proof is complete.
The constant is well defined, but its value has to be estimated. From a numerical computation we conclude that .
When solving (2.2) numerically we need Cauchy data , along the line . The most natural way to obtain this is to use two thermocouples, located at , and , and compute by solving a well-posed problem for the Helmholtz equation in the interval . Hence it is natural to assume knowledge about at a second point.
These two results can be combined into an error estimate for the spectral method. This is demonstrated in two examples.
Note that, under these assumptions, can be used as a rule for selecting the regularization parameter.
By balancing these two components and , we can find a suitable value for the regularization parameter.
3. Numerical Implementation
The matrix approximates the second-order derivative by spectral cutoff method (see ). This ordinary differential equation system can easily be solved by various numerical methods.
It is easy to verify that the function is the exact solution of (1.2). Now we need to seek the solution , where from the Cauchy data pairs and .
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).
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