International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 943621, 20 pages

doi:10.1155/2012/943621

Research Article

A Regularization of the Backward Problem for Nonlinear Parabolic Equation with Time-Dependent Coefficient

Pham Hoang Quan,¹ Le Minh Triet,¹ and Dang Duc Trong²

¹Department of Mathematics and Applications, Sai Gon University, 273 An Duong Vuong, District 5, Ho Chi Minh City, Vietnam

²Department of Mathematics, University of Natural Science, Vietnam National University, 227 Nguyen Van Cu, District 5, Ho Chi Minh City, Vietnam

Received 31 March 2012; Revised 21 June 2012; Accepted 11 August 2012

Academic Editor: Theodore E. Simos

Copyright © 2012 Pham Hoang Quan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the backward problem with time-dependent coefficient which is a severely ill-posed problem. We regularize this problem by combining quasi-boundary value method and quasi-reversibility method and then obtain sharp error estimate between the exact solution and the regularized solution. A numerical experiment is given in order to illustrate our results.

1. Introduction

We consider the inverse time problem for the nonlinear parabolic equation

where is thermal conductivity function of (1.1) such that there exist ,   satisfying

for all .

In other words, from the temperature distribution at a particular time (final data), we want to retrieve the temperature distribution at any earlier time . This problem is called the backward heat problem (BHP), or final-value problem. As known, this problem is severely ill-posed in Hadamard’s sense; that is, solutions do not always exist, and when they exist, they do not depend continuously on the given data. In practice, datum is based on (physical) measurements. Hence, there will be measurement errors, and we would actually have datum function such that . Thus, form small error contaminating physical measurements, the solutions corresponding to datum function have large errors. This makes it difficult to make numerical calculations with perturbed data.

In our knowledge, there have been many papers on the linear homogeneous case of the backward problem, but there are a few papers on the nonhomogeneous case and the nonlinear case such as [1–6]; especially, the nonlinear case with time-dependent thermal conductivity coefficient is very scarce. Moreover, the thermal conductivity is the property of a material’s ability to conduct heat. Therefore, the thermal conductivity is not a constant in some cases. In this paper, we extend the result (in [7]) for the case of the time-dependent thermal conductivity . In future, we will research the BHP problem for the case of the time- and space-dependent thermal conductivity .

In [8], the authors used the quasi-reversibility method to regularize a 1D linear nonhomogeneous backward problem. Very recently, in [9], the methods of integral equations and of Fourier transform have been used to solve a 1D problem in an unbounded region. In recent articles considering the nonlinear backward heat problem, we refer the reader to [10]. In [11], the authors used the quasi-boundary value method to regularize the latter problem. However, in [11], the authors showed that the error between the regularized solution and the exact solution is

It is easy to see that the convergence of the error estimate between the regularized solution and the exact solution is very slow when is in a neighborhood of zero. For this reason, the error estimate in initial time is given by

We can easily find that the exact solution of (1.1)–(1.3) satisfies

where

In this paper, we will approximate (1.1)–(1.3) by using the regularization problem:

where . Actually, in [12], we also considered the problem (1.1)–(1.3) for the homogeneous case in . Hence, we want to extend for the nonlinear case in bounded region , and this is the biggest different point in this paper.

The remainder of this paper is organized as follows. In Section 2, we shall regularize the ill-posed problem (1.1)–(1.3) and give the error estimate between the regularized solution and the exact solution. Then, in Section 3, a numerical example is given.

2. Regularization and Error Estimates

For clarity of notation, we denote the solution of (1.1)–(1.3) by and the solution of the regularized problem (1.12) by . Throughout this paper, we denote , and, be a positive number such that . Hereafter, we have a number of inequalities in order to evaluate error estimates.

Lemma 2.1. Let be a function satisfying (1.4), ,   and be as in (1.11). Then for and , one gets

(i), (ii).

Proof of Lemma 2.1. The proof of Lemma 2.1 can be found in [12].

Theorem 2.2. Let and such that there exists independent of satisfying

Then problem (1.12) has a unique weak solution satisfying the following equality:

where

Proof of Theorem 2.2. Step 1. The existence and the uniqueness of the solution of the problem (2.2)

Put

where

We claim that, for every ,  , we have

where and is supremum norm in . We shall prove this inequality by induction. For , we have

From Lemma 2.1, we get

It follows

Therefore, we have

where .

Thus, (2.6) holds for . Supposing that (2.6) holds for , we shall prove that (2.6) holds for . In fact, we get

Hence, we obtain

Thus, we have

Therefore, by the induction principle, we have

for all .

We consider . Since

when there exists a positive integer number such that

and is a contraction. It follows that the equation has a unique solution .We claim that . In fact, one has . Hence, . By the uniqueness of the fixed point of , one has ; that is, the equation has a unique solution .

Step 2. If satisfies (2.2), then is the solution of the problem (1.12). For , we have

We can verify directly that . In fact, . Moreover, one has

Hence, is the solution of (1.12).

Step 3. The problem (1.12) has at most one solution . In fact, let and be two solutions of (1.12) such that . Putting , then satisfies

It follows that

By using the result in Lees and Protter [13], we get . This completes the proof of Step 3.

Finally, by combining three steps, we complete the proof of Theorem 2.2.

Theorem 2.3 (stability of the modified method). Let f be as in Theorem 2.2, and let in satisfy . If one supposes that and defined by (2.2) are corresponding to the final values and in , respectively, then one obtains

Proof of Theorem 2.3. Using the inequality and Lemma 2.1, we get the estimate

Thus, we get

Hence, we obtain

Thus, we get

By using Gronwall’s inequality, we have

It follows

This completes the proof of Theorem 2.3.

Theorem 2.4. Let u be the exact solution of problem (1.1)–(1.3) such that

for all . Letting and given by (2.2) corresponding to the perturbed data , then one has for every

where .

Proof of Theorem 2.4. From (1.1), we construct the regularized solution corresponding to the exact data and the perturbed data

where

Since (1.8) and (2.32), we get

From , we get

By applying the inequality , we get

Using Lemma 2.1, we obtain

where

Hence, we get the following estimates

From the estimate (2.39), we get

Hence, we have

Applying Gronwall’s inequality, we obtain

Hence, we get

From (1.1), we have

where

Therefore, we get the estimate

Let be the solution of (1.12) corresponding to the perturbed data , and let be the solution of (1.12) corresponding to the exact data . From Theorem 2.3 and (2.46), we can get

where .

This completes the proof of Theorem 2.4.

3. Numerical Experiment

Consider the nonlinear parabolic equation with time-dependent coefficient:

where

The exact solution of the equation is

Letting , from (3.3), we have

Consider the measured data

Then we have

From (2.31) and (3.5), we have the regularized solution for the case in the form of iteration

where

We consider ,  ,  ,  ,  ,  ,  ,  , and . Now, we get Table 1 for the case .

Table 1

We have in Figure 1 the graphs of the regularized solution ,   and .

(a)

(b)

(c)

Figure 1: The regularized solutions corresponding to ,  .

We have in Figure 2 the graphs of the regularized solution ,   and .

(a)

(b)

(c)

Figure 2: The regularized solutions corresponding to ,  .

We have in Figure 3 the graphs of the exact solution and of the regularized solution ,   and .

(a)

(b)

(c)

Figure 3: The regularized solutions corresponding ,   and the exact solution.

Now, Figure 4 can represent visually the exact solution and regularized solutions corresponding to ,   at initially time .

Figure 4: The exact solution and the regularized solutions corresponding to ,   at initial time .

Notice that, in Figure 4, the curve number 0 expressing the exact solution is indistinguishable from the curve number expressing the regularized solution corresponding to ,  .

Remark 3.1. From (1.7) and (3.5), we obtain the exact solution corresponding to the measured data :

where

Now, we cannot calculate the formula (3.9) exactly (we need to find while we have not known yet). From Theorem 2.2, we use the iteration for (3.9) at initial time as follows:

where

Then we get the error in the Table 2.

We can see that the error is very large. Therefore, the problem is ill-posed and a regularization is necessary.

Table 2

Acknowledgment

All authors were supported by the National Foundation for Science and Technology Development (NAFOSTED).

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