We consider a backward heat conduction problem (BHCP) with variable coefficient. This problem is severely ill-posed in the sense of Hadamard and the regularization techniques are required to stabilize numerical computations. We use an iterative method based on the truncated technique to treat it. Under an a-priori and an a-posteriori stopping rule for the iterative step number, the convergence estimates are established. Some numerical results show that this method is stable and feasible.
In this article, we consider the following backward heat conduction problem (BHCP) with variable coefficient
where
and
our purpose is to determine
This problem is severely ill-posed and the regularization techniques are required to stabilize numerical computations [
Followed the work in [
Inspired by [
This paper is constructed as follows. In Section 2, we make a simple review for the ill-posedness of problem (1) and give the description of our iteration method. Section 3 is devoted to the convergence estimates under two stopping rules. Numerical results are shown in Section 4. Some conclusions are given in Section 5.
We make a simple review for the ill-posedness of problem (1) (also see [
We denote
Further, we suppose that the corresponding eigenfunctions
then the eigenfunctions
From [
where
Setting
from (5) and the integration formula by parts, we know
thus, the solution (6) can be rewritten as
From (9), it can be observed that
to recovery the stability of solution
In this subsection, we give our iteration method. Firstly, given
this is a direct problem, use the similar method as in [
Now, for
then, for
Take
Let the exact and noisy data
where
and we note that
Now, we truncate (16) to obtain the following our iterative algorithm
where N is a positive constant, which plays a role of the regularization parameter.
For simplicity, we take the initial guess as zero, then our iterative scheme becomes
Further, we suppose that there exists a constant
In the iterative process, the iterative step number k can be chosen by the a-priori and a-posteriori rules. In this subsection, we choose it by an a-priori rule and give the convergence estimate for the iterative algorithm.
Theorem 3.1. Suppose that u given by (6) is the exact solution of problem (1) with the exact data
Proof. For
where
Use the triangle inequality, it is clear that
From the Equations (6), (19) with the exact data
On the other hand, from the Equation (19) with the exact and measured data
From the above estimates of
In the iterative process, the a-priori stopping rule
For the iterative scheme (19), we control the iterative step number k by the following form
where
Theorem 3.2. Suppose that u given by (6) is the exact solution of problem (1) with the exact data
Proof. Firstly, for the estimate of
Below, we estimate
then, we get
Now, from the Equations (6), (19) with the exact data
From the above estimates of
Remark 3.3.
For the a-priori case, in problem (1) and the inequality (2), if we take
then it can be obtained that
Note that,
where
Similarly, for the a-posteriori case, we can derived the convergence result of order optimal
where
In this section, we use a numerical example to verify how this method works. Since the ill-posedness for the case at
Example. We take
where
As in (10), (11), the solution of problem (32) can be written as
here,
and the measured data
In addition, we define the relative root mean square errors (RRMSE) between the exact and approximate solution is given by
In order to make the convenient and accurate computation, we adopt the a-posteriori stopping rule (26) to choose the iterative step k. During the computation procedure, we take
For
0.0001 | 0.001 | 0.005 | 0.01 | 0.1 | |
---|---|---|---|---|---|
0.00019 | 0.0019 | 0.0095 | 0.0185 | 0.1905 | |
k | 160.0000 | 118.0000 | 89.0000 | 77.0000 | 34.0000 |
From
An iterative method is based on the truncated technique to solve a BHCP with variable coefficients. Under an a- priori and an a-posteriori selection rule for the iterative step number, the convergence estimates are established. Some numerical results show that this method is stable and feasible.
The authors appreciate the careful work of the anonymous referee and the suggestions that helped to improve the paper. The work is supported by the the SRF (2014XYZ08), NFPBP (2014QZP02) of Beifang University of Nationalities, the SRP of Ningxia Higher School (NGY20140149) and SRP of State Ethnic Affairs Commission of China (14BFZ004).
Hongwu Zhang,Xiaoju Zhang, (2015) Iterative Method Based on the Truncated Technique for Backward Heat Conduction Problem with Variable Coefficient. Open Access Library Journal,02,1-11. doi: 10.4236/oalib.1101501