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A Cauchy problem for the elliptic equation with variable coefficients is considered. This problem is severely ill-posed. Then, we need use the regularization techniques to overcome its ill-posedness and get a stable numerical solution. In this paper, we use a modified Tikhonov regularization method to treat it. Under the a-priori bound assumptions for the exact solution, the convergence estimates of this method are established. Numerical results show that our method works well.

In this paper, we consider the following Cauchy problem for the elliptic equation with variable coefficients in a strip region

where

Without loss of generality, in the following section we suppose that

Let

In 2007, Hào et al. [

In this article, we continue to consider the problem (1). We adopt a modified Tikhonov regularization method to solve it. Under the a-priori bound assumptions for the exact solution, we give and proof the convergence estimates for this method. It can be seen that the convergence result is order optimal [

This paper is constructed as follows. In Section 2, we give some auxiliary results for this paper. In Section 3, we make the description for modified Tikhonov regularization method, and Section 4 is devoted to the convergence estimates for this method. Numerical results and some conclusions are shown in Sections 5-6, respectively.

For a function

Firstly, we consider the following Cauchy problem in the frequency domain

Lemma 2.1 [

(i)

(ii)

(iii)

(iv) there exist positive constants

here

Secondly, Take the Fourier transform of problem (1) with respect to

It can be shown that, for

then, the exact solution of problem (1) can be expressed by

Note that

Further, we suppose that there exists a constant

or

here

We firstly give the description for this method. Note that, from (9), we have

According to (15), for

and

Let the exact and noisy datum

where

Denote

By Theorem 2.11 of Chapter 2 in [

According to Parseval equality, we get

thus,

and

from (20), we have

Combing with (22), (23), (24), we can obtain that

hence,

using the inverse Fourier transform, we get the following Tikhonov regularization solution for problem (1)

Note that, the above Tikhonov regularization solution (27) can be interpreted as using the regularized kernel

Now, we choose the regularization parameter by the a-priori rule and give the convergence estimates for this method.

Theorem 4.1 Suppose that

where,

Proof. From (10), (28), (18), (12), we have

According to Lemma 2.1, one can obtain that

Set

Let

for

and note that,

thus, we get

From (34), we can derive that

combing with (36), (37), we have

Consequently,

Now we estimate

adopting the similar proof procedure, we have

and

Hence,

From the selection of regularization parameter

Theorem 4.1 shows that, for the fixed

Theorem 4.2 Suppose that

Proof. From (10), (28), (18), (13) and (14), we have

By Lemma 2.1, we can know

using the similar derivation processes with

then from (45) and the selection rule

Below, we estimate

Case 1: for the large values with

Case 2: for

Then, by (50), (51), we can obtain that

Consequently, from the selection rule

Remark 4.3 From the convergence estimate (44), we can see that the logarithmic term with respect to

the dominating term. Asymptotically this yields a convergence rate of order

In this section, a numerical example is given to verify the stability and efficiency of our proposed method.

Taking

We use the discrete Fourier transform (DFT) and inverse Fourier transform (IFT) to complete our numerical experiments. The exact and regularized solutions are computed by (10) and (28), respectively. For

In order to make a comprehensive analysis for the convergence with respect to the error level

and the corresponding computation results are shown in

From Figures 1(a)-(d) and

A Cauchy problem for the elliptic equation with variable coefficients is considered. We use the modified Tikhonov regularization method to overcome its ill-posedness. Convergence estimates of this method are esta- blished under the a-priori selections for regularization parameter. Some numerical results show that our method works well.

0.0001 | 0.0005 | 0.001 | 0.005 | 0.01 | |
---|---|---|---|---|---|

0.0026 | 0.0075 | 0.0137 | 0.0519 | 0.0653 |

The author would like to thank the reviewers for their constructive comments and valuable suggestions that improve the quality of our paper. The work described in this paper was supported by the NSF of China (11371181) and the SRF of Beifang University of Nationalities (2014XYZ08).

HongwuZhang, (2014) Modified Tikhonov Method for Cauchy Problem of Elliptic Equation with Variable Coefficients. American Journal of Computational Mathematics,04,213-222. doi: 10.4236/ajcm.2014.43018