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Publisher: Taylor & Francis

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Inverse Problems in Science and Engineering

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A modified quasi-boundary value method for regularizing of a backward problem with time-dependent

coefficient

Pham Hoang Quan a , Dang Duc Trong b , Le Minh Triet a & Nguyen Huy Tuan a

a

Department of Mathematics and Applications , Saigon University , 273 An Duong Vuong, Ho Chi Minh City , Vietnam

b

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

Published online: 21 Apr 2011

To cite this article: Pham Hoang Quan , Dang Duc Trong , Le Minh Triet & Nguyen Huy Tuan

(2011) A modified quasi-boundary value method for regularizing of a backward problem with

time-dependent coefficient, Inverse Problems in Science and Engineering, 19:3, 409-423, DOI: 10.1080/17415977.2011.552111

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Inverse Problems in Science and Engineering

Vol 19, No 3, April 2011, 409–423

A modified quasi-boundary value method for regularizing of a backward

problem with time-dependent coefficient

Pham Hoang Quana*, Dang Duc Trongb

, Le Minh Trietaand Nguyen Huy Tuanay

a

Department of Mathematics and Applications, Saigon University, 273 An Duong Vuong,

Vietnam National University, 227 Nguyen Van Cu, Q.5, Ho Chi Minh City, Vietnam

(Received 5 June 2010; final version received 31 December 2010)

In this article, we consider a classical ill-posed problem which is the backward problem for parabolic equation with time-dependent coefficient

uxxðx, tÞ ¼ aðtÞutðx, tÞ, ðx, tÞ 2 R  ½0, T :

Applying the method of Fourier transform and the modified quasi-boundary value method, we shall give a regularization for the problem A sharp error estimate between the regularized solution and the exact solution is given in this article A numerical experiment is provided illustrating the main results

Keywords: backward problem; Fourier transform; ill-posed problems; parabolic equation; time-dependent coefficient

AMS Subject Classifications: 34K29; 35K05; 42A38; 37L65

1 Introduction

In this article, we consider the backward problem for parabolic equation with time-dependent coefficient In fact, we shall find the temperature u(x, t) satisfying

uxxðx, tÞ ¼ aðtÞutðx, tÞ, ðx, tÞ 2 R  ½0, T , ð1Þ

uðx, T Þ ¼ gðxÞ, x 2 R, ð2Þ where a(t), g(x) are given functions such that a(t) 4 0 Note that the system (1), (2) can be solved directly or we can set a variable FðtÞ ¼Rt

0 1 aðsÞds then we shall get the system of equations with constant coefficients by simple calculation However, if the system (1), (2) become the system of nonhomogeneous equations

uxxðx, tÞ  aðtÞutðx, tÞ ¼ f ðx, tÞ, ðx, tÞ 2 R  ½0, T ,

uðx, T Þ ¼ gðxÞ, x 2 R,

*Corresponding author Email: quan.ph@cb.sgu.edu.vn

Vietnam

ISSN 1741–5977 print/ISSN 1741–5985 online

ß 2011 Taylor & Francis

DOI: 10.1080/17415977.2011.552111

the problem will be more complicated than (1), (2) In our opinion, it is difficult to use variable F(t) in the case of the system of nonhomogeneous equations Therefore, we choose the direct method to solve homogeneous equations and mention some results in the case of nonhomogeneous equations (Theorem 3)

As is known, the problem (1), (2) is usually ill-posed in Hadamard’s sense, i.e the existence does not always hold, and in the case of existence, the solution does not depend continuously on the given data Thus, an appropriate regularization method is required The backward problem for the parabolic equation has been studied using many methods in the last four decades There has been much research work in this area, such as [1,5–9] The latter problem is a special case of the general one of finding u satisfying

utþAðtÞu ¼0, t 2 ½0, T , uðT Þ ¼ g,

where A(t) is a linear operator in an appropriate function space In [2], Lattes–Lions used the quasi-reversibility (QR) method for regularizing the main equation by adding the

‘stability term’ in to the main equation In 1973, Miller dealt with the problem by using the

‘stability term’ f(A)

utþf ðAÞu ¼0, t 2 ½0, T , uðT Þ ¼ g:

Another method which was called the quasi-boundary value (QBV) method was researched by many authors Using this method, they add the ‘stability term’ into the boundary value For example, in [10], Denche and Bessila used this method to regularize the backward problem for the parabolic equation with

utþf ðAÞu ¼0, t 2 ½0, T , uðT Þ  u0ð0Þ ¼ ’:

Very recently, in [4], the authors used the quasi-boundary method to regularize a backward heat problem and obtained an error estimate which is of order Tt at t 6¼ 0 and an error estimate which is of order ðlnð1

ÞÞ1 at zero It is easy to show that if t is in a neighbourhood of zero, the convergence of the approximate solution is very slow From some disadvantages of the error estimate which is of order Tt, we shall improve the error estimate in order to speed up the convergence of the approximate solution for every

t 2[0, T )

In our knowledge, papers related to the problem with time-dependent operator A(t) are scarce In this article, we consider the special case in which

AðtÞu ¼  1

aðtÞuxx: The problem (1), (2) is ill posed Hence, a regularization is in order In fact, by taking the Fourier transform (with respect to x) two sides of (1) and using the condition (2), we obtain the exact solution

uðx, tÞ ¼ 1ffiffiffiffiffiffi

2

p

Zþ1

e!2ðFðT ÞFðtÞÞbgð!Þei!xd!, ð3Þ

bgð!Þ ¼ 1ffiffiffiffiffiffi

2

p

Zþ1

1

gðxÞei!xdx, ð4Þ

FðtÞ ¼

Zt

0

1

In this article, to get a stable approximation of u we shall use

uðgÞð!, tÞ ¼ 1ffiffiffiffiffiffi

2

p

Zþ1

1

e! 2 FðtÞ

!2þe! 2 FðT Þbgð!Þei!xd!, ð6Þ and

vðgÞð!, tÞ ¼ 1ffiffiffiffiffiffi

2

p

Zþ1

1

e!2ðFðtÞþmÞ

!2þe! 2 ðFðT ÞþmÞbgð!Þei!xd!: ð7Þ

In fact, if we apply the QBV method [11] for regularizing (1), (2), we shall get the regularized problem

rxxðx, tÞ ¼ aðtÞr

tðx, tÞ, ðx, tÞ 2 R  ½0, T ,

rðx, T Þ þ rðx, 0Þ ¼ gðxÞ, x 2 R:

Applying the Fourier method, we can get the solution of the above problem

rðgÞð!, tÞ ¼ 1ffiffiffiffiffiffi

2

p

Zþ1

1

e!2FðtÞ

 þ e! 2 FðT Þbgð!Þei!xd!:

It is easy to see that the regularized parameter of ris  which is different from the regularized parameter of the regularized solution (6), (7) Moreover, the form of the regularized solution (7) is completely different from r Under some strong conditions on the smoothness of the exact data gex, we will show that

uðgÞð, tÞ  uexð, tÞ

2T0ð1 þ ffiffiffiffi

Q

p ÞFðT ÞFðtÞ½lnðFðT Þ=ÞFðtÞFðT ÞFðT Þ , and

vðgÞð, tÞ  uexð, tÞ

2Tmð1 þ ffiffiffiffiffiffiffi

Qm

p ÞFðT ÞþmFðtÞþm½lnðFðT Þ=ÞFðtÞFðT ÞFðT Þþm, where

T0¼maxf1, FðT Þg, 0 5 Q ¼

Z

R

!2e!2FðT Þbgexð!Þ



 2

d! 5 1

and

Tm¼maxf1, FðT Þ þ mg, 0 5 Qm¼

Z

R

!2e!2ðFðT ÞþmÞbgexð!Þ



 2

d! 5 1:

The remainder of this article is divided into two sections In Section 2, we shall prove some regularization results In Section 3, a numerical experiment is given

Inverse Problems in Science and Engineering 411

2 Regularization results

2.1 The stability of the regularized solution

LEMMA 1 Let0 5  5 M and gðÞ ¼þe1M Then we get

gðÞ  M

ð1 þ lnðM=ÞÞ

M

lnðM=Þ for every  0

Proof The proof of Lemma 1 can be found in [12]

LEMMA 2 Let 0  t  s  M, 0 5  5 M,  2 R and M1¼max{1, M}, we can get the following inequalities

(i) e

ðstMÞ 2

2þeM 2 M1½lnðM=ÞtsM, (ii) e

t 2

2þeM 2 M1½lnðM=ÞtMM : Proof

(i) We get

eðstMÞ 2

2þeM 2 eðstMÞ 2

ð2þeM 2

ÞstMð2þeM 2

ðeM 2

ÞMs t M

:

Thus, we obtain

eðstMÞ 2

2þeM 2 M

lnðM=Þ

 s

M1½lnðM=ÞtsM, where M1¼max{1, M}

(ii) Let s ¼ M, we get et2

 2 þe M2 M1½lnðM=ÞtMM: This completes the proof of Lemma 2

LEMMA 3 (The stability of the regularized solution given by (6)) Let  2(0, F(T )),

g1, g22L2(R) and u(g1), u(g2) be two solutions given by (6) corresponding to the final values

g1, g2, respectively Then we obtain

uðg1Þð, tÞ  uðg2Þð, tÞ

2T0½lnðFðT Þ=ÞFðtÞFðT ÞFðT Þ g1g2

2, where kk2is the norm in L2(R) and T0¼max{1, F(T )}

Proof From (6) and Lemma 2, we have

buðg1Þð!, tÞ buðg2Þð!, tÞ

 ¼ e! 2 FðtÞ

!2þe! 2 FðT Þðgb1ð!Þ gb2ð!ÞÞ













T0½lnðFðT Þ=ÞFðtÞFðT ÞFðT Þ ðgb1ð!Þ gb2ð!ÞÞ:

Hence, we get

buðg1Þð, tÞ buðg2Þð, tÞ

2T0½lnðFðT Þ=ÞFðtÞFðT ÞFðT Þ gb1gb2

2: From Plancherel’s Theorem, we obtain

uðg1Þð, tÞ  uðg2Þð, tÞ

2T0½lnðFðT Þ=ÞFðtÞFðT ÞFðT Þ g1g2

2: This completes the proof of Lemma 3

LEMMA 4 (The stability of the regularized solution given by (7)) Let  2(0, F(T )), m 4 0,

g1, g22L2(R) and v(g1), v(g2) be two solutions given by (7) corresponding to the final values

g1, g2, respectively Then we obtain

vðg1Þð, tÞ  vðg2Þð, tÞ

2Tm½lnððFðT Þ þ mÞ=ÞFðtÞFðT ÞFðT Þþmg1g2

2, where kk2is the norm in L2(R) and Tm¼max{1, F(T ) þ m}

Proof From (7) and Lemma 2, we have

bvðg1Þð!, tÞ bvðg2Þð!, tÞ

 ¼ e!

2 ðFðtÞþmÞ

!2þe! 2 ðFðT ÞþmÞðgb1ð!Þ gb2ð!ÞÞ













 Tm½lnððFðT Þ þ mÞ=ÞFðtÞFðT ÞFðT Þþmðgb1ð!Þ gb2ð!ÞÞ



Hence, we get

bvðg1Þð, tÞ bvðg2Þð, tÞ

2Tm½lnððFðT Þ þ mÞ=ÞFðtÞFðT ÞFðT Þþm gb1gb2

2: From Plancherel’s Theorem, we obtain

vðg1Þð, tÞ  vðg2Þð, tÞ

2Tm½lnððFðT Þ þ mÞ=ÞFðtÞFðT ÞFðT Þþm g1g2

2: This completes the proof of Lemma 4

2.2 Regularization of (1), (2)

Suppose that uex(!, t) is the exact solution of the backward problem for the parabolic equation corresponding to the exact data gex, and g is the measured data satisfying

kg"gexk2"where kk2is the norm in L2(R)

From (6), we shall construct the regularized solution corresponding to the measured data

u"ðg"Þð!, tÞ ¼ 1ffiffiffiffiffiffi

2

p

Zþ1

1

e!2FðtÞ

!2þe! 2 FðT Þgb"ð!Þei!xd!, ð8Þ and the regularized solution corresponding to the exact data

u"ðgexÞð!, tÞ ¼ 1ffiffiffiffiffiffi

2

p

Zþ1

e! 2 FðtÞ

!2þe! 2 FðT Þgcexð!Þei!xd!: ð9Þ Inverse Problems in Science and Engineering 413

We get the estimate

u"ðg"Þð, tÞ  uexð, tÞ

2¼bu"ðg"Þð, tÞ buexð, tÞ

2

bu"ðg"Þð, tÞ bu"ðgexÞð, tÞ

2þbu"ðgexÞð, tÞ buexð, tÞ

2 ð10Þ

THEOREM 1 Suppose that  2(0, F(T )), uex(, t) 2 L2(R) 8t 2 [0, T ), g, gex2L2(R) such that

kg"gexk2" and0 5 Q ¼R

Rj!2e! 2 FðT Þbgexð!Þj2d! 5 1: Then we have

uðgÞð, tÞ  uexð, tÞ

2 T0ð1 þ ffiffiffiffi

Q

p ÞFðT ÞFðtÞ½lnðFðT Þ=ÞFðtÞFðT ÞFðT Þ , for every t 2[0, T ], where T0¼max{1, F(T )}

Proof From Lemma 3, we get

buðgÞð, tÞ buðgexÞð, tÞ

2T0½lnðFðT Þ=ÞFðtÞFðT ÞFðT Þ ggex

2

From (3) and (9), we have

buðgexÞð!, tÞ buexð!, tÞ

 ¼ e! 2 FðtÞ

!2þe! 2 FðT Þgcexð!Þ  e

! 2 FðtÞ

e! 2 FðT Þgcexð!Þ













¼ !2e!2FðT Þ

!2þe! 2 FðT Þ

!

e!2FðtÞgcexð!Þ













¼ !2e! 2 FðtÞ

!2þe! 2 FðT Þ

!

e!2FðT Þgcexð!Þ















 ,

in which T0¼max{1, F(T )}

Thus, we obtain

buðgexÞð, tÞ buexð, tÞ

Z

R

!2e!2FðT Þbgexð!Þ



 2

d!

¼T0 ffiffiffiffi Q

p

where 0 5 Q ¼R

Rj!2e! 2 FðT Þbgexð!Þj2d! 5 1:

From (10), (11) and (12), we get

buðgÞð, tÞ buexð, tÞ

2T0ð1 þ ffiffiffiffi

Q

p ÞFðT ÞFðtÞ½lnðFðT Þ=ÞFðtÞFðT ÞFðT Þ : Thus, from Plancherel’s Theorem, we obtain

uðgÞð, tÞ  uexð, tÞ

2T0ð1 þ ffiffiffiffi

Q

p ÞFðT ÞFðtÞ½lnðFðT Þ=ÞFðtÞFðT ÞFðT Þ : This completes the proof of Theorem 1

THEOREM 2 Suppose that  2(0, F(T )), uex(, t) 2 L2(R) 8t 2 [0, T ), g, gex2L2(R) such that

kg g k " and v be the solution defined by (7) If we assume, in addition, that there

exists a positive number m such that 0 5 Qm¼R

Rj!2e! 2 ðFðT ÞþmÞbgexð!Þj2d! 5 1: Then

we have

vðgÞð, tÞ  uexð, tÞ

2Tmð1 þ ffiffiffiffiffiffiffi

Qm

p ÞFðT ÞþmFðtÞþm½lnðFðT Þ=ÞFðtÞFðT ÞFðT Þþm

for every t 2[0, T ], where Tm¼max{1, F(T ) þ m}

Proof From (7), we construct the regularized solution corresponding to the measured data

vðgÞð!, tÞ ¼ 1ffiffiffiffiffiffi

2

p

Zþ1

1

e!2ðFðtÞþmÞ

!2þe! 2 ðFðT ÞþmÞgbð!Þei!xd!, ð13Þ and the regularized solution corresponding to the exact data

vðgexÞð!, tÞ ¼ 1ffiffiffiffiffiffi

2

p

Zþ1

1

e! 2 ðFðtÞþmÞ

!2þe! 2 ðFðT ÞþmÞgcexð!Þei!xd!: ð14Þ

We get the estimate

v"ðg"Þð, tÞ  uexð, tÞ

2¼bv"ðg"Þð, tÞ buexð, tÞ

2

bv"ðg"Þð, tÞ bv"ðgexÞð, tÞ

2þbv"ðgexÞð, tÞ buexð, tÞ

2: ð15Þ From Lemma 4, we have

bv"ðg"Þð, tÞ bv"ðgexÞð, tÞ

2Tm ln FðT Þ þ m



FðT Þþm

ggex

2



FðT Þþm

, ð16Þ

where Tm¼max{1, F(T ) þ m}

On the other hand, we get

bvðgexÞð!, tÞ buexð!, tÞ

 ¼ e!

2 ðFðtÞþmÞ

!2þe! 2 ðFðT ÞþmÞgcexð!Þ  e

! 2 ðFðtÞþmÞ

e! 2 ðFðT ÞþmÞgcexð!Þ













¼ e! 2 ðFðtÞþmÞe! 2 ðFðT ÞþmÞe! 2 ðFðtÞþmÞð!2þe! 2 ðFðT ÞþmÞÞ

e! 2 ðFðT ÞþmÞð!2þe! 2 ðFðT ÞþmÞÞ gcexð!Þ













e! 2 ðFðT ÞþmÞð!2þe! 2 ðFðT ÞþmÞÞe!2ðFðtÞþmÞgcexð!Þ



¼ e!2ðFðtÞþmÞ ð!2þe! 2 ðFðT ÞþmÞÞ!2e!2ðFðT ÞþmÞgcexð!Þ











:

From Lemma 2, we have

bvðgexÞð!, tÞ buexð!, tÞ

   T m½lnððFðT Þ þ mÞ=ÞFðtÞFðT ÞFðT Þþm!2e!2ðFðT ÞþmÞgcexð!Þ

Inverse Problems in Science and Engineering 415

Thus, we obtain

bvðgexÞð, tÞ buexð:, tÞ

2Tm ffiffiffiffiffiffiffi

Qm

p

in which 0 5 Qm¼R

Rj!2e! 2 ðFðT ÞþmÞbgexð!Þj2d! 5 1:

From (15)–(17), we have

vðgÞð, tÞ  uexð, tÞ

2Tmð1 þ ffiffiffiffiffiffiffi

Qm

p

This completes the proof of Theorem 2

Remark

(1) We see that the rate of convergence of error estimates in Theorems 1 and 2 is better than in [4] because of the improved rate of convergence in the neighbourhood of zero In particular, the rate of convergence at initial time t ¼ 0 (in Theorem 2) is faster than other regularization schemes (in Theorem 1 or in [4])

(2) In Theorems 1 and 2, we got the error estimate in the case of homogeneous equations More generally, we consider the system of nonhomogeneous equations

wxxðx, tÞ  aðtÞwtðx, tÞ ¼ f ðx, tÞ, ðx, tÞ 2 R  ½0, T , ð19Þ

wðx, T Þ ¼ gðxÞ, x 2 R: ð20Þ

By applying the Fourier transform, we can get the exact solution of (19), (20)

b wð!, tÞ ¼ e!2ðFðT ÞFðtÞÞgcexð!Þ þ

ZT

t

e!2½FðsÞFðtÞbfð!, sÞ

aðsÞ ds: ð21Þ

In Theorem 3, we suggest the regularized solution of (19), (20) corresponding to the exact data gexand the measured data g

b

wðgexÞð!, tÞ ¼ e

! 2 FðtÞ

!2þe! 2 FðT Þgcexð!Þ þ

ZT

t

e! 2 ½FðsÞFðtÞFðT Þ

!2þe! 2 FðT Þ

bfð!, sÞ aðsÞ ds, ð22Þ

b

wðgÞð!, tÞ ¼ e

! 2 FðtÞ

!2þe! 2 FðT Þgbð!Þ þ

ZT

t

e! 2 ½FðsÞFðtÞFðT Þ

!2þe! 2 FðT Þ

bfð!, sÞ aðsÞ ds: ð23Þ Then we shall obtain the error estimate (24)

THEOREM 3 Suppose that  2(0, F(T )), g, gex2L2(R) such that kg"gexk2" and w be the exact solution of(19), (20) satisfying w(., t) 2 L2(R) 8t 2 [0, T ), wxx(, 0) 2 L2(R) and

0 5 K1¼ ffiffiffi

2

p

wxxð, 0Þ

 2

2þT Z

R

ZT

0

!2e!2FðsÞbfð!, sÞ

aðsÞ













2

ds d!

5 1:

If we assume that w(g) is defined by (23) then we have

wðgÞð, tÞ  wð, tÞ

2T0ð1 þ K1ÞFðT ÞFðtÞ½lnðFðT Þ=ÞFðtÞFðT ÞFðT Þ ð24Þ for every t 2[0, T ], where T ¼max{1, F(T )}