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