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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