DSpace at VNU: On a backward heat problem with time-dependent coefficient: Regularization and error estimates

This problem is ill-posed in the sense that the solution if it exists does not depend continuously on the final data.. This is usually referred to as the backward heat conduction problem,

On a backward heat problem with time-dependent coefficient:

Regularization and error estimates

a

Department of Applied Mathematics, Faculty of Science and Technology, Hoa Sen University, Quang Trung Software Park, Dist 12, Ho Chi Minh City, Viet Nam

b

Department of Mathematics and Applications, SaiGon University, 273 An Duong Vuong st., Dist 5, Ho Chi Minh City, Viet Nam

c

Department of Mathematics, University of Natural Science, Vietnam National University, 227 Nguyen Van Cu st., Dist 5, Ho Chi Minh City, Viet Nam

a r t i c l e i n f o

Keywords:

Backward problem

Fourier transform

Ill-posed problems

Heat equation

Time-dependent coefficient

a b s t r a c t

In this paper, we consider a homogeneous backward heat conduction problem which appears in some applied subjects This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the final data A new regularization method

is applied to formulate regularized solutions which are stably convergent to the exact ones with Holder estimates A numerical example shows that the computational effect of the method is all satisfactory

Ó 2012 Elsevier Inc All rights reserved

1 Introduction

There are several important ill-posed problems for parabolic equations A classical example is the backward heat equa-tion In other words, it may be possible to specify the temperature distribution at a particular time t < T from the temper-ature data at the final time t ¼ T This is usually referred to as the backward heat conduction problem, or the final value problem In the present paper, we consider the problem of finding the temperature uðx; tÞ; ðx; tÞ 2 ½0;p  ½0; T such that

where aðtÞ; gðxÞ are given The problem is called the backward heat problem, (BHP for short), the backward Cauchy problem

or the final value problem In general, the solution of the problem does not exist Further, even if the solution existed, it would not be continuously dependent on the final data It makes difficult to do numerical calculations Hence, a regulariza-tion is in order

In the special case of the problem(1)–(3)with aðtÞ ¼ 1, the problem becomes

The problem(4)–(6)has been considered by many authors using different methods The mollification method has been studied in [4] An iterative algorithm with regularization techniques has been developed to approximate the BHP by Jourhmane and Mera in[10] Kirkup and Wadsworth have given an operator-splitting method in[9] Quasi-reversibility

0096-3003/$ - see front matter Ó 2012 Elsevier Inc All rights reserved.

⇑ Corresponding author.

E-mail address: lmtriet2005@yahoo.com.vn (T Le Minh).

Contents lists available atSciVerse ScienceDirect

Applied Mathematics and Computation

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / a m c

method has been used by Lattes and Lions[1], Miller[2]and the other authors[3,13,11] The boundary element method has been also used by some authors (see[6,8]) All of them were devoted to computational aspects However, few authors gave their error estimates from the theoretical viewpoint for the BHP except Schroter and Tautenhahn[5], Yildiz and Ozdemir[7]

and Yildiz et al.[12]

Although we have many works on (4)–(6), however to the author’s knowledge, so far there are few results about

(1)–(3) The major object of this paper is to provide a regularization method to establish the Holder estimates In fact,

we decided to regularize the exact problem by using the form of(13)directly It can be called the quasi-solution method (but it based on the quasi-boundary value method) By using quasi boundary value method, we have the regularized problem as follows

By applying the Fourier method, we can find the form of the solution of(7)–(9)

uðx; tÞ ¼X1

m¼1

exp m2Rt

0aðsÞds

bþ exp m2RT

0aðsÞds

The regularized solution(13)based on modifying the solution(10)of the problem(7)–(9)(noting that whena¼ 0, the solution(13)is the solution(10)) In this paper, we use the regularized solution(13)directly InTheorem 2.2, we can get the error estimate of Holder type for all t by using an appropriate parameteraP0 In fact, the error estimate for the case

0 < t < T is as follows

uð; tÞ vð; tÞ

k k 6 ð1 þ A1Þp2 tþpq2 Tþqaa:

In this case, we can choosea¼ 0 and require a soft condition of the exact solution u

A1¼ uð; 0Þk k < 1:

On the other hand, the error estimate for the case t ¼ 0 is as follows

uð; tÞ vð; tÞ

k k 6 ð1 þ A1Þq2 Tþqpaa:

In order to get the the error estimate of Holder type, we choosea>0 and require a strong condition of the exact solution u

A1¼ p

2

X1

m¼1

exp 2m 2a

umð0Þ

!1

<1:

It requires the exact solution u is smooth enough The remainder of the paper is divided into two sections In Section2, we establish the regularized solution and estimate the error between an exact solution u of problem(1)–(3)and the regularized solution uwith the Holder type Finally, a numerical experiment will be given in Section3

2 Regularization and main results

We denote that k  k is the norm in L2ð0;pÞ Let h; i be the inner product in L2ð0;pÞ and gbe the measured data satisfying

kgðÞ  gðÞk 6 Let aðtÞ : ½0; T ! R be the differentiable function for every t and satisfy

Suppose that Problems(1)–(3)have an exact solution u then u can be formulated as follows

uðx; tÞ ¼X1

m¼1

exp m2Rt

0aðsÞds

exp m2RT

0aðsÞds

Let b > 0 andaP0, we shall approximate the solution of the backward heat problem(1)–(3)by the regularized solution

as follows

vðx; tÞ ¼X1

m¼1

exp m2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

We note that b depends onesuch that lim

e !0bðeÞ ¼ 0 andais an arbitrarily nonnegative number

Next, we consider some lemmas which is useful to the proof of theorems

Lemma 2.1 Let x 2 R;c>0, 0 6 a 6 b, and b–0 then

exa

1 þcexb6c a

Proof of Lemma 2.1 We have

exa

xa

ð1 þcexbÞað1 þcexbÞ1a

ð1 þcexbÞa

6c a

:

This completes the proof ofLemma 2.1 h

Lemma 2.2 Let aðtÞ satisfy(11)and 0 < b < 1 Then for m P 1, one has

exp m2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

for allaP0

Proof of Lemma 2.2 FromLemma 2.1, we obtain

exp m2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

b

 cðtÞ

where cðtÞ ¼

RT

0 aðsÞdsRt

0 aðsÞds

RT

0 aðsÞdsþ a

From(11), we get

Z T

0

aðsÞds P

Z T 0

pds ¼ pT;

Z T

t

aðsÞds 6

Z T t

qds ¼ qðT  tÞ:

Hence we get cðtÞ 6qðTtÞ

pTþ a Thus, since (15) gives

exp m2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

b

 qðTtÞ pTþ a

¼ bqðtTÞpTþ a:

This completes the proof ofLemma 2.2 h

Lemma 2.3 Let aðtÞ satisfy(11)and 0 < b < 1 Then for m P 1, one has

bexp m2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

for allaP0

Proof of Lemma 2.3 From (15), we have

bexp m2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

b

 cðtÞ

¼ b1cðtÞ:

From(11), we get

bexp m2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

This completes the proof of Lemma2.3 h

Theorem 2.1 Let 0 < b < 1 andaP0 If v and w in Y are defined by(13)corresponding to the final values g and h in L2

ð0;pÞ, respectively then

vð; tÞ  wð; tÞ

Proof of Theorem 2.1 Fromvand w are defined by(13)corresponding to the final values g and h in L2ð0;pÞ, we have

vðx; tÞ ¼X1

m¼1

exp m2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

and

wðx; tÞ ¼X1

m¼1

exp m2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

where

gm¼2

p

Z p

0

gðxÞ sinðmxÞdx; hm¼2

p

Z p

0

By usingLemma 2.2, we obtain

vð; tÞ  wð; tÞ

2

X1 m¼1

exp m2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa













2

2b

2qðtTÞ pTþ aX1 m¼1

gm hm

j j2¼ b2qðtTÞpTþ akg  hk2: ð20Þ

Therefore, we get

vð; tÞ  wð; tÞ

This completes the proof ofTheorem 2.1 h

Theorem 2.2 Let b ¼p;aP0 and g; g2 L2ð0;pÞ satisfy g  gk k 6 If we suppose that u is the solution of problem(1)–(3)

such that

A1¼ p

2

X1

m¼1

exp 2m 2a

umð0Þ

!1

then one has for every t 2 ½0; T

uð; tÞ vð; tÞ

wherevðx; tÞ is defined by(13)corresponding to the noisy data gðxÞ

Proof of Theorem 2.2 Let ube defined by(13)with exact data g Using the triangle inequality, we get

uð; tÞ vð; tÞ

For the term uk ð; tÞ vð; tÞk, using(16), we estimate it as follows

uð; tÞ vð; tÞ

From(12), we get the mth Fourier sine coefficient of u

umðtÞ ¼

exp m2Rt

0aðsÞds

exp m2RT

0aðsÞds

Since(13), we get

mðtÞ ¼

exp m2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

From(26) and (27)and usingLemma 2.3, we obtain

umðtÞ  umðtÞ

2Rt

0aðsÞds

exp m2RT

0aðsÞds

2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

0

@

1

Agm













2 Rt

0aðsÞds þa

exp m2 RT

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

2 Rt

0aðsÞds þa

bþ exp m2 RT

0aðsÞds þa

umð0Þ

j j 6 bqTþptþaaexp m 2a

umð0Þ

This follows that

uð; tÞ  uð; tÞ

2

X1 m¼1

umðtÞ  umðtÞ

6b2ðptþqTþ aaÞp 2

X1 m¼1

exp 2m 2a

umð0Þ

j j2:

Hence, we obtain

uð; tÞ  uð; tÞ

where A2

¼p

2

P1

m¼1exp 2m 2a

umð0Þ

j j2 From(24), (25),(29)and b ¼p, we have

uð; tÞ v

ð; tÞ

k k 6 A1bptþqTþaaþ bqðtTÞpTþ a6A1 p

ptþ a qTþ a

þ p

qðtTÞ pTþ a

6A1p2 tþpq2 TþqaaþptþqTþaa6ð1 þ A1Þq2 Tþqp2 tþpaa:

This completes the proof ofTheorem 2.2 h

3 Numerical experiment

Consider the linear homogeneous parabolic equation with time-dependent coefficient

utðx; tÞ ¼ aðtÞuxxðx; tÞ; ðx; tÞ 2 ½0;p  ð0; 1;

uð0; tÞ ¼ uðp;tÞ ¼ 0; t 2 ½0; 1;

where

and

uðx; 1Þ ¼ gexðxÞ ¼sin x

Hence, we obtain

gex

Z p

0

sin x

e2



 2ds

!1

¼

ffiffiffiffi p 2

r

e2

and

1 6 aðtÞ 6 3

for all t 2 ½0; 1

The exact solution of the equation is

uðx; tÞ ¼X1

m¼1

exp m2Rt

0aðsÞds

exp m2R1

0aðsÞds

where gm¼2

p

Rp

0gexðxÞ sinðmxÞdx

Let t ¼ 0, from(32), we have

uðx; 0Þ ¼X1

m¼1

1

e2m 2gmsinðmxÞ:

Consider the measured data

geðxÞ ¼ 1 þ e

ffiffiffi

p

2

p

e2

!

then we have

gegex e

ffiffiffi

p

2

pe2

Z p

0

sin x

e2



 

2

dx

!1

¼e:

Leta¼ 0, from(13) and (33), we have the regularized solution for the case t ¼ 0

veðx; 0Þ ¼X1

m¼1

1

bþ e2m 2gmesinðmxÞ;

where gme¼2

p

Rp

0geðxÞ sinðmxÞdx anda¼ 0

Letebee1¼ 101;e2¼ 102;e3¼ 103;e4¼ 104;e5¼ 105and b ¼e1

, respectively Let

ReðtÞ ¼ vei ð;tÞuð;tÞ

uð; tÞ

be the relative error between the exact solution and the regularized solution at the time t

Then we shall make the comparison between the absolute error and the relative error in the case t ¼ 0 and t ¼ 0:1 We have the following table for the case t ¼ 0

e1¼ 101 6.411579e001 5.1157575999e001

e2¼ 102 5.914401e001 4.7190616771e001

e3¼ 103 4.215352e001 3.3634022181e001

e4¼ 104 2.549425e001 2.0341697917e001

e5¼ 105 1.372796e001 1.0953450889e001

We have the following table for the case t ¼ 0:1

e1¼ 101 5.743711e001 5.1155245814e001

e2¼ 102 5.298322e001 4.7188475240e001

e3¼ 103 3.776257e001 3.3632499109e001

e4¼ 104 2.283862e001 2.0340773067e001

e5¼ 105 1.229797e001 1.0952947987e001

where uk exð; 0Þk ¼ ð2ð1=2Þ pð1=2ÞÞ=2 ’ 1:2533 and uk exð; 0Þk ¼pð1=2Þ  ð1=ð2  expð11=50ÞÞÞð1=2Þ ’ 1:1228

We have the following graph of the exact solution uexð; tÞ and of the regularized solutionve ið; tÞ; i ¼ 1; 2:

We have the following graph of the regularized solutionve ið; tÞ; i ¼ 3; 4; 5:

Now, the figure can represent visually the exact solution and the regularized solution at initally time t = 0 Now, the figure can represent visually the exact solution and the regularized solution at the time t = 0.1

All authors were supported by the National Foundation for Science and Technology Development (NAFOSTED) We thank the referees for constructive comments leading to the improved version of the paper

References

[1] R Lattes, J.L Lions, Methode de Quasi-reversibilite et Applications, Dunod, Paris, 1967.

[2] K Miller, Stabilized quasi-reversibility and other nearly-best-possible methods for non-well posed problems, in: Symposium on Non-Well Posed Problems and Logarithmic Convexity, Lecture Notes in Mathematics, vol 316, Springer-Verlag, Berlin, 1973 pp 161–176.

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[9] S.M Kirkup, M Wadsworth, Solution of inverse diffusion problems by operator-splitting methods, Appl Math Modell 26 (2002) 1003–1018 [10] M Jourhmane, N.S Mera, An iterative algorithm for the backward heat conduction problem based on variable relaxation factors, Inv Prob Eng 10 (2002) 293–308.

[11] I.V Mel’nikova, Q Zheng, J Zheng, Regularization of weakly ill-posed Cauchy problem, J Inv Ill-posed Prob 10 (5) (2002) 385–393.

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