^{1}

^{*}

^{1}

A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven; a convergence estimate of Hölder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution; some numerical results show that this method works well.

Let

where

Further, we suppose

there exists a nontrivial solution

Our problem is to determine

Problem (1.1) is severely ill-posed, i.e., a small perturbation in the given Cauchy data may result in a dramatic error on the solution [

In the present paper, we adopt a filtering function method to deal with this problem. The idea of this method is similar to the ones in [

This paper is organized as follows. In Section 2, we use the filtering function method to treat problem (1.1) and prove some well-posed results (the existence, uniqueness and stability for the regularization solution). In Section 3, a Hölder type convergence estimate for the regularized method is derived under an a-priori bound assumption for the exact solution. Numerical results are shown in Section 4. Some conclusions are given in Section 5.

We assume there exists a solution to problem (1.1), then it satisfies the following nonlinear integral equation (see [

here,

From (2.1), we can see that the functions

so in order to guarantee the convergence of solution

frequencies of

called filtering function method.

Let

where

filter function

where,

In fact, it can be verified that (2.4) satisfies the following mixed boundary value problem formally

Our idea is to approximate the exact solution (2.1) by the regularization solution (2.4), i.e., using the solution of (2.5) to approximate the one of (1.1).

Let

then

note that, when

Now, we prove that the problem (2.4) is well-posed (existence, uniqueness and stability for the regularization solution), the proof mentality of Theorem 2.1 mainly comes from the references [

Theorem 2.1. Let

Proof. For

then for

where

For

Note that, for

(2.9), (1.2), we have

When

then for

By the induction principle, we can obtain that

hence, it is clear that

We consider

There must exist a positive integer number

it shows that the equation

In the following, we give and prove the stability of the regularization solution.

Theorem 2.2 Suppose f satisfies (1.2),

measured datum

where

Proof. From (2.4), we have

where

By (2.17), (2.18), (2.7), (2.8) and (1.2), we have

Subsequently,

using Gronwall’s inequality [

then from the above inequality (2.19), the stability result (2.16) can be obtained. □

In this section, under an a-priori bound assumption for the exact solution a convergence estimate of Hölder type for the regularization method is derived. The corresponding result is shown in Theorem 3.1.

Theorem 3.1. Suppose that f satisfies the uniform Lipschitz condition (1.2), and u given by (2.1) is the exact solution of problem (1.1),

and the regularization parameter

then for fixed

here

Proof. Denote

From Theorem 2.2, for

By (2.1), (2.4), (2.7), (2.8), we have

For

use Gronwall’s inequality [

thus

From (3.2), (3.4), (3.5), (3.7) and (2.3), we can obtain the convergence result (3.3). □

In this section, we verify the accuracy and efficiency of our given regularization method by the following numerical example

here we take

It is clear that

data as

Let

here

For a fixed

We adopt the above given algorithms to compute the regularization solution at

for

From

0.00001 | 0.0001 | 0.001 | 0.01 | 0.05 | ||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 8303e−06 | 1 | 8303e−05 | 1 | 8303e−04 | 0 | 0018 | 0 | 0092 | |

0 | 0087 | 0 | 0088 | 0 | 0094 | 0 | 0284 | 0 | 1036 | |

0 | 0094 | 0 | 0095 | 0 | 0105 | 0 | 0290 | 0 | 1111 |

We use a filtering function method to solve a Cauchy problem for semi-linear elliptic equation. The results of the well-posedness for the approximation problem are given. Under the a-priori bound assumption, the conver- gence estimate of Hölder type has been derived. Finally, we compute the regularization solution by constructing an iterative scheme. Some numerical results show that this method is stable and feasible.

The authors 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 SRF (2014XYZ08, 2015JBK423), NFPBP (2014QZP02) of Beifang University of Nationalities, the SRP of Ningxia Higher School (NGY20140149) and SRP of State Ethnic Affairs Commission of China (14BFZ004).

HongwuZhang,XiaojuZhang, (2015) Filtering Function Method for the Cauchy Problem of a Semi-Linear Elliptic Equation. Journal of Applied Mathematics and Physics,03,1599-1609. doi: 10.4236/jamp.2015.312184