Filtering Function Method for the Cauchy Problem of a Semi-Linear Elliptic Equation ()
Received 3 November 2015; accepted 14 December 2015; published 17 December 2015
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1. Introduction
Let
be a bounded, connected domain in
with a smooth boundary
and assume that H is a real Hilbert space. We consider the following Cauchy problem of a semi-linear elliptic partial differential equation
(1.1)
where
denotes a linear densely defined self-adjoint and positive-definite operator with respect to x. The function
is known, and
is an uniform Lipschitz continuous function, i.e., existing
independent of
,
,
such that
(1.2)
Further, we suppose
be the eigenvalues of the operator
, i.e., for the boundary value problem
(1.3)
there exists a nontrivial solution
. And
satisfy
(1.4)
Our problem is to determine
from problem (1.1).
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 [1] . Thus regularization techniques are required to stabilize numerical computations, (see [1] [2] ). We know that, as the right term
, it is the Cauchy problem of the homogeneous elliptic equations. For the homogeneous problem, there have many regularization methods to deal with it, (see [3] -[8] ). We note that, these references mainly consider the Cauchy problem of linear homogeneous elliptic operator equation, but the literature which involves the semi-linear cases is quite scarce. In 2014, [9] considered the problem (1.1), where the authors used Fourier truncated method to solve it and derived the convergence estimate of logarithmic type. Recently, there are some similar works about the Cauchy problem for nonlinear elliptic equation, and they have been published, such as [10] [11] .
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 [4] [5] [12] [13] , etc. However, note that our method here is new and different from them in the above references (see Section 2). Meanwhile we will derive the convergence estimate of Hölder type for this method, which is an improvement for the result in [9] .
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.
2. Filtering Function Method and Some Well-Posed Results
2.1. Filtering Function Method
We assume there exists a solution to problem (1.1), then it satisfies the following nonlinear integral equation (see [9] )
(2.1)
here,
are the orthonormal eigenfunctions for the operator
, and
(2.2)
is the inner product in H.
From (2.1), we can see that the functions
,
tend to infinity (as
),
so in order to guarantee the convergence of solution
, the high frequencies(
) of two functions need to be eliminated. Therefore, a natural way is to use a filter function
to filter out the high
frequencies of
,
and obtain a stable approximate solution, this is so-
called filtering function method.
Let
be the noisy data, and satisfying
(2.3)
where
is the error level,
is the H-norm. According to the above description, for
, we choose the
filter function
, and define the following regularization solution
(2.4)
where,
,
.
In fact, it can be verified that (2.4) satisfies the following mixed boundary value problem formally
(2.5)
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).
2.2. Some Well-Posed Results
Let
,
, for the fixed
, we define the function
(2.6)
then
attain unique maximum at the point
, and from
,
, we have
(2.7)
note that, when
, it can be obtained that
(2.8)
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 [14] , which describes the ex- istence and uniqueness for the solution of (2.4).
Theorem 2.1. Let
, f satisfies (1.2), then the problem (2.4) exists a unique solution
.
Proof. For
, we consider the operator
defined by
(2.9)
then for
,
, we can prove the following estimate is valid
(2.10)
where
,
denotes the sup norm in
.
For
, we firstly use the induction principle to prove
(2.11)
Note that, for
, from (2.7),
. Meanwhile, use the basic inequalities
,
, and
. When
, from
(2.9), (1.2), we have
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When
, we suppose
(2.12)
then for
, by (2.12), it similarly can be proven that
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By the induction principle, we can obtain that
(2.13)
hence, it is clear that
(2.14)
We consider
, and from real analysis, we know
(2.15)
There must exist a positive integer number
, such that
, therefore
is a contraction,
it shows that the equation
has a unique solution
. Noting that
, thus,
. By the uniqueness of the fixed point of
, we have
, so the equation
has a unique solution
. □
In the following, we give and prove the stability of the regularization solution.
Theorem 2.2 Suppose f satisfies (1.2),
and
be the solutions of problem (2.4) corresponding to the
measured datum
and
, respectively, then for
, we have
(2.16)
where
.
Proof. From (2.4), we have
(2.17)
(2.18)
where
,
.
By (2.17), (2.18), (2.7), (2.8) and (1.2), we have
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Subsequently,
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using Gronwall’s inequality [15] , we have
(2.19)
then from the above inequality (2.19), the stability result (2.16) can be obtained. □
3. Convergence Estimate
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),
defined by (2.4) is the regularization solution, the measured data
satisfies (2.3). If the exact solution u satisfies
(3.1)
and the regularization parameter
is chosen as
(3.2)
then for fixed
, we have the following convergence estimate
(3.3)
here
,
,
is given in Theorem 2.2.
Proof. Denote
be the solution of problem (2.4) with exact data
. We know that
(3.4)
From Theorem 2.2, for
, we have
(3.5)
By (2.1), (2.4), (2.7), (2.8), we have
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For
, we get
(3.6)
use Gronwall’s inequality [15] , it can be obtained that
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thus
(3.7)
From (3.2), (3.4), (3.5), (3.7) and (2.3), we can obtain the convergence result (3.3). □
4. Numerical Experiments
In this section, we verify the accuracy and efficiency of our given regularization method by the following numerical example
(4.1)
here we take
,
,
, then
and
.
It is clear that
is an exact solution of problem (4.1), thus
,
. We choose the measured
data as
, where
is an error level, and
(4.2)
Let
for
, the regularization solution
with
can be computed by the following iteration scheme
(4.3)
here
, and
(4.4)
(4.5)
For a fixed
, in order to make the sensitivity analysis for numerical results, we define the relative root mean square error between the exact and approximate solutions as
(4.6)
We adopt the above given algorithms to compute the regularization solution at
with
,
for
Taking
for
the numerical results for
and
at
,
are shown in Figure 1 and Figure 2, respectively. For
, the relative root mean square errors for the various error levels
and regularization parameters
at
are shown in Table 1. In the computational procedure, the regulari- zation parameter
is chosen by (3.2), and
is computed by (4.2).
From Figure 1 and Figure 2 and Table 1, it can be observed that our regularization method is effective and stable. Meanwhile we note that the smaller
is, the better the calculation effect is. Table 1 shows that the numerical results become worse when y approaches to 1, which is a common phenomenon in the computation of ill-posed Cauchy problems for the elliptic equation.
5. Conclusion
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.
Acknowledgements
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).
NOTES
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*Corresponding author.