DSpace at VNU: A modified quasi-boundary value method for regularizing of a backward problem with time-dependent coefficient

This article was downloaded by: [Universite Laval] On: 16 July 2014, At: 01:13 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Inverse Problems in Science and Engineering Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gipe20 A modified quasi-boundary value method for regularizing of a backward problem with time-dependent coefficient a b a Pham Hoang Quan , Dang Duc Trong , Le Minh Triet & 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 To link to this article: http://dx.doi.org/10.1080/17415977.2011.552111 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content Downloaded by [Universite Laval] at 01:13 16 July 2014 This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions 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 Trieta and Nguyen Huy Tuanay a Department of Mathematics and Applications, Saigon University, 273 An Duong Vuong, Ho Chi Minh City, Vietnam; bDepartment of Mathematics, University of Natural Science, Vietnam National University, 227 Nguyen Van Cu, Q.5, Ho Chi Minh City, Vietnam Downloaded by [Universite Laval] at 01:13 16 July 2014 (Received 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 uxx x, tị ẳ atịut x, tị, x, tị R ẵ0, 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 Introduction In this article, we consider the backward problem for parabolic equation with timedependent coefficient In fact, we shall find the temperature u(x, t) satisfying uxx ðx, tị ẳ atịut x, tị, x, tị R ẵ0, T , ux, T ị ẳ gxị, x R, ð1Þ ð2Þ where a(t), g(x) are given functions such that a(t)R4 Note that the system (1), (2) can be t ds then we shall get the system of solved directly or we can set a variable FðtÞ ¼ aðsÞ equations with constant coefficients by simple calculation However, if the system (1), (2) become the system of nonhomogeneous equations uxx x, tị atịut x, tị ẳ f x, tị, ux, T ị ẳ gxị, x, tị R  ½0, T Š, x R, *Corresponding author Email: quan.ph@cb.sgu.edu.vn y Current affiliation: Institute for Computational Science and Technology, Ho Chi Minh City, Vietnam ISSN 1741–5977 print/ISSN 1741–5985 online ß 2011 Taylor & Francis DOI: 10.1080/17415977.2011.552111 http://www.informaworld.com 410 P.H Quan et al 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 ut ỵ Atịu ẳ 0, t ½0, T Š, Downloaded by [Universite Laval] at 01:13 16 July 2014 uT ị ẳ 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) ut ỵ f Aịu ẳ 0, t ẵ0, T , uT ị ẳ 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 ut ỵ f Aịu ẳ 0, uT ị u0 0ị ẳ : t ½0, T Š, Very recently, in [4], the authors used the quasi-boundary method to regularize a t backward heat problem and obtained an error estimate which is of order T at t 6¼ and an error estimate which is of order ðlnð1ÞÞÀ4 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 t some disadvantages of the error estimate which is of order T , we shall improve the error estimate in order to speed up the convergence of the approximate solution for every t [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 Atịu ẳ uxx : atị 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 Z ỵ1 e! FT ịFtịịb 3ị g!ịei!x d!, ux, tị ẳ p 2 À1 Inverse Problems in Science and Engineering 411 where b g!ị ẳ p 2 Z ỵ1 gxịei!x dx, 4ị Z t Ftị ẳ ds: asị ð5Þ In this article, to get a stable approximation of u we shall use u gị!, tị ẳ p 2 Z ỵ1 e! Ftị i!x b g!ịe d!, ! ỵ e!2 FT ị 6ị Downloaded by [Universite Laval] at 01:13 16 July 2014 and v gị!, tị ẳ p 2 Z ỵ1 e! Ftịỵmị i!x b g!ịe d!: !2 ỵ e!2 FT ịỵmị 7ị In fact, if we apply the QBV method [11] for regularizing (1), (2), we shall get the regularized problem rxx x, tị ẳ atịrt x, tị, x, tị R ẵ0, T , r x, T ị ỵ rx, 0ị ẳ gxị, x R: Applying the Fourier method, we can get the solution of the above problem r gị!, tị ẳ p 2 Z ỵ1 e! Ftị i!x b g!ịe d!:  ỵ e!2 FT ị It is easy to see that the regularized parameter of r is  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 pffiffiffiffi FðtÞ FðtÞÀFðT Þ u ð g ÞðÁ, tÞ uex , tị T0 ỵ QịFT ị ẵlnFT Þ=ފ FðT Þ , and v ð g ị, tị uex , tị Tm ỵ p Ftịỵm FtịFT ị Qm ịFT ịỵm ẵlnFT ị=ị FT ịỵm , where T0 ẳ maxf1, FT ịg, 05Q ẳ Z !2 FðT Þ b gex ð!Þ d! ! e R and Tm ¼ maxf1, FT ị ỵ mg, Qm ẳ Z !2 FT ịỵmị b gex !ị d! 1: ! e R 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 412 P.H Quan et al Regularization results 2.1 The stability of the regularized solution LEMMA Let  M and gị ẳ ỵe1M Then we get gị M 1 ỵ lnM=ịị M  lnM=ị for every  ! Proof The proof of Lemma can be found in [12] LEMMA Let t following inequalities s M,  M,  R and M1 ¼ max{1, M}, we can get the Downloaded by [Universite Laval] at 01:13 16 July 2014 estMị (i)  ỵ eM2 et (ii)  ỵ eM2 Proof ts M1 ẵ lnM=ị M , tM M1 ẵ lnM=ị M : (i) We get estMị 2 ỵ eM2 estMị Mỵts M st 2 ỵ eM2 ị M 2 ỵ eM2 ị estMịx Mỵts M eM2 ị s t 2 ỵ eM2 ịMM : Thus, we obtain estMị 2 ỵ eM2  M  lnM=ị Ms Mt ts M1 ẵ lnM=ị M , where M1 ẳ max{1, M} t2 (ii) Let s ẳ M, we get 2eỵeM2 tM M1 ẵ lnðM=ފ M : This completes the proof of Lemma LEMMA (The stability of the regularized solution given by (6)) Let  (0, F(T )), g1, g2 L2(R) and u(g1), u(g2) be two solutions given by (6) corresponding to the final values g1, g2, respectively Then we obtain FðtÞÀFðT Þ u ð g1 ÞðÁ, tÞ À u g2 ị, tị T0 ẵ lnFT ị=ị FT Þ g1 À g2 , 2 where kÁk2 is the norm in L2(R) and T0 ¼ max{1, F(T )} Proof From (6) and Lemma 2, we have eÀ! FðtÞ b b u ð g1 Þð!, tị b u g2 ị!, tị ẳ b g !ị g !ịị ! ỵ eÀ!2 FðT Þ FðtÞÀFðT Þ g1 ð!Þ À gb2 !ịị: T0 ẵ lnFT ị=ị FT ị b Inverse Problems in Science and Engineering 413 Hence, we get b u ð g2 ÞðÁ, tÞ 2 u ð g1 ÞðÁ, tị b T0 ẵ lnFT ị=ị FtịFT ị FT Þ gb1 À gb2 : From Plancherel’s Theorem, we obtain u ð g1 ÞðÁ, tÞ À u g2 ị, tị T0 ẵ lnFT ị=ị FtịFT Þ FðT Þ g1 À g2 : This completes the proof of Lemma LEMMA (The stability of the regularized solution given by (7)) Let  (0, F(T )), m 0, g1, g2 L2(R) and v(g1), v(g2) be two solutions given by (7) corresponding to the final values g1, g2, respectively Then we obtain FtịFT ị Tm ẵ lnFT ị ỵ mị=ị FT ịỵm g1 À g2 , Downloaded by [Universite Laval] at 01:13 16 July 2014 v ð g1 ÞðÁ, tÞ À v ð g2 ÞðÁ, tÞ where kÁk2 is the norm in L2(R) and Tm ¼ max{1, F(T ) þ m} Proof From (7) and Lemma 2, we have !2 Ftịỵmị e b b v g2 Þð!, tÞ ¼ ðb g ð!Þ À g ð!ÞÞ v g1 ị!, tị b ! ỵ e!2 FT ịỵmị FtịFT ị g1 !ị gb2 !ịị: Tm ẵ lnFT ị ỵ mị=ị FT ịỵm b Hence, we get FtịFT ị Tm ẵ lnFT ị ỵ mị=ị FT ịỵm gb1 gb2 : b v ð g1 ÞðÁ, tÞ À b v ð g2 ÞðÁ, tÞ 2 From Plancherel’s Theorem, we obtain FðtÞÀFðT ị Tm ẵ lnFT ị ỵ mị=ị FT ịỵm g1 À g2 : v ð g1 ÞðÁ, tÞ À v ð g2 ÞðÁ, tÞ This completes the proof of Lemma 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" À gexk2 " where kÁk2 is the norm in L2(R) From (6), we shall construct the regularized solution corresponding to the measured data u" g" ị!, tị ẳ p 2 Z ỵ1 e! Ftị gb" !ịei!x d!, !2 ỵ e!2 FT ị 8ị and the regularized solution corresponding to the exact data u" ð gex Þð!, tị ẳ p 2 Z ỵ1 e! Ftị i!x d!: gc ex !ịe ! ỵ e!2 FT Þ ð9Þ 414 P.H Quan et al We get the estimate u" ð g" ÞðÁ, tÞ À uex , tị ẳ b u" g" ị, tị b uex ðÁ, tÞ 2 b u" ð g" ÞðÁ, tÞ À b u" ð gex ÞðÁ, tÞ 2 þ b u" ð gex ÞðÁ, tÞ À b uex ðÁ, tÞ 2 ð10Þ THEOREM Suppose that  )), uex(Á, t) L2(R) 8t [0, T ), g, gex L2(R) such that R (0, F(T !2 FðT Þb gex ð!Þj2 d! 1: Then we have kg" À gexk2 " and Q ¼ R j! e u ð g ÞðÁ, tÞ À uex , tị T0 ỵ p Ftị FtịFT ị QịFT ị ẵlnFT ị=ị FT ị , for every t [0, T ], where T0 ¼ max{1, F(T )} Proof From Lemma 3, we get Downloaded by [Universite Laval] at 01:13 16 July 2014 b u ð gex ÞðÁ, tÞ 2 u ð g ÞðÁ, tÞ À b T0 ½ lnðFðT Þ=ފ FðtÞ FðT Þ T0  FðtÞÀFðT Þ FT ị ẵlnFT ị=ị g gex FtịFT ị FðT Þ : ð11Þ From (3) and (9), we have À!2 FðtÞ À!2 FðtÞ e e b c c u ð gex Þð!, tÞ À b uex !, tị ẳ !ị !ị g g ex ! ỵ e!2 FT ị ex e!2 FT ị ! !2 e! FðT Þ À!2 FðtÞ c ẳ !ị g e ex !2 ỵ eÀ! FðT Þ ! !2 eÀ!2 FðtÞ !2 FT ị c ẳ !ị e g ex !2 ỵ e!2 FT ị Ftị FtịFT ị T0 FT ị ẵlnFT ị=ị FT ị !2 e! FT Þ gc ex ð!Þ, in which T0 ¼ max{1, F(T )} Thus, we obtain b u ð gex ÞðÁ, tÞ À b uex ðÁ, tÞ 2 FðtÞ FðT Þ T0  ẵlnFT ị=ị FtịFT ị FT ị Z 1=2 !2 FðT Þ b gex ð!Þ d! ! e R p Ftị FtịFT ị ẳ T0 QFT ị ½lnðFðT Þ=ފ FðT Þ , R ð12Þ where Q ẳ R j!2 e! FT ịb gex !ịj2 d! 1: From (10), (11) and (12), we get b uex ðÁ, tÞ 2 u ð g ÞðÁ, tÞ b T0 ỵ p Ftị FtịFT ị QịFT Þ ½lnðFðT Þ=ފ FðT Þ : Thus, from Plancherel’s Theorem, we obtain pffiffiffiffi FðtÞ FðtÞÀFðT Þ u ð g ị, tị uex , tị T0 ỵ QịFT Þ ½lnðFðT Þ=ފ FðT Þ : This completes the proof of Theorem THEOREM Suppose that  (0, F(T )), uex(Á, t) L2(R) 8t [0, T ), g, gex L2(R) such that kg" À gexk2 " and v be the solution defined by (7) If we assume, in addition, that there Inverse Problems in Science and Engineering 415 R exists a positive number m such that Qm ¼ R j!2 e! FT ịỵmịb gex !ịj2 d! 1: Then we have p Ftịỵm FtịFT ị v g ị, tị uex , tị Tm ỵ Qm ịFT ịỵm ẵlnFT ị=ị FT ịỵm for every t [0, T ], where Tm ẳ max{1, F(T ) ỵ m} Proof data From (7), we construct the regularized solution corresponding to the measured v g ị!, tị ẳ p 2 Z ỵ1 e! Ftịỵmị i!x gb !ịe d!, !2 ỵ e!2 FT ịỵmị 13ị Downloaded by [Universite Laval] at 01:13 16 July 2014 and the regularized solution corresponding to the exact data v ð gex ị!, tị ẳ p 2 Z ỵ1 e! Ftịỵmị i!x d!: gc ex !ịe ! ỵ e!2 FT ịỵmị 14ị We get the estimate v" g" ị, tị uex , tị ẳ b v" ð g" ÞðÁ, tÞ À b uex ðÁ, tÞ 2 b v" ð g" ÞðÁ, tÞ À b v" gex ị, tị 2 ỵ b v" gex ÞðÁ, tÞ À b uex ðÁ, tÞ 2 : ð15Þ From Lemma 4, we have b v" ð g" ÞðÁ, tÞ À b v" ð gex ÞðÁ, tÞ 2   FtịFT ị FT ị ỵ m FT ịỵm g À gex Tm  ln  Þ   FtịFT Ftịỵm FT ị ỵ m FT ịỵm Tm FT ịỵm ln ,  16ị where Tm ẳ max{1, F(T ) ỵ m} On the other hand, we get !2 Ftịỵmị !2 Ftịỵmị e e b c c uex !, tị ẳ !ị !ị g g v ð gex Þð!, tÞ À b ex ex 2 ! FT ịỵmị ! ỵ e! FT ịỵmị e e! Ftịỵmị e!2 FT ịỵmị e!2 Ftịỵmị !2 ỵ e!2 FT ịỵmị ị c ẳ g !ị ex e!2 FT ịỵmị !2 ỵ e!2 FT ịỵmị ị ! !2 Ftịỵmị ẳ !2 FT ịỵmị gc ex !ị FT ịỵmị e ! e ! ỵ e ị e! Ftịỵmị !2 FT ịỵmị ẳ gc ex !ị: FT ịỵmị ! e ! ! ỵ e Þ From Lemma 2, we have b v ð gex Þð!, tÞ À b uex ð!, tÞ FtịFT ị Tm ẵ lnFT ị ỵ mị=ị FT ịỵm !2 e! FT ịỵmị gc ex !ị Ftịỵm FtịFT ị ẳ Tm FT ịỵm ẵlnFT ị ỵ mị=ị FT ịỵm !2 e! FT ịỵmị gc ex ð!Þ: 416 P.H Quan et al Thus, we obtain p Ftịỵm FtịFT ị b v gex ị, tị b uex :, tị 2 Tm Qm FT ịỵm ẵlnFT ị ỵ mị=ị FT ịỵm , R in which Qm ẳ R j!2 e! FT ịỵmịb gex ð!Þj2 d! 1: From (15)–(17), we have pffiffiffiffiffiffiffi Ftịỵm FtịFT ị v g ị, tị uex , tị Tm ỵ Qm ịFT ịỵm ẵlnFT ị ỵ mị=ị FT ịỵm : 17ị 18ị This completes the proof of Theorem Downloaded by [Universite Laval] at 01:13 16 July 2014 Remark (1) We see that the rate of convergence of error estimates in Theorems and 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 (in Theorem 2) is faster than other regularization schemes (in Theorem or in [4]) (2) In Theorems and 2, we got the error estimate in the case of homogeneous equations More generally, we consider the system of nonhomogeneous equations wxx x, tị atịwt x, tị ẳ f x, tị, x, tị R ẵ0, T , wx, T ị ẳ gxị, x R: 19ị ð20Þ By applying the Fourier transform, we can get the exact solution of (19), (20) b!, tị ẳ e w !2 FT ịFtịị gc ex !ị ỵ Z T e! ẵFsịFtị t b f!, sị ds: asị 21ị In Theorem 3, we suggest the regularized solution of (19), (20) corresponding to the exact data gex and the measured data g b gex ị!, tị ẳ w e! Ftị gc ex !ị ỵ ! ỵ e!2 FT Þ eÀ! FðtÞ b ð g Þð!, tÞ ¼ w gb !ị ỵ ! ỵ e!2 FT ị Z T !2 ẵFsịFtịFT ị b t Z t f!, sÞ e ds, FðT Þ À! aðsÞ ! ỵ e T !2 ẵFsịFtịFT ị b f!, sị e ds: !2 ỵ e!2 FT ị asị 22ị 23ị Then we shall obtain the error estimate (24) THEOREM Suppose that  (0, F(T )), g, gex L2(R) such that kg" À gexk2 " and w be the exact solution of (19), (20) satisfying w(., t) L2(R) 8t [0, T ), wxx(Á, 0) L2(R) and " #12 Z Z T b pffiffiffi fð!, sÞ ! FðsÞ K1 ẳ wxx , 0ị 2 ỵT ds d! 1: ! e aðsÞ R If we assume that w(g) is defined by (23) then we have FðtÞ FðtÞÀFðT Þ w ð g ÞðÁ, tÞ À wðÁ, tÞ T0 ỵ K1 ịFT ị ẵlnFT ị=ị FT ị for every t [0, T ], where T0 ¼ max{1, F(T )} ð24Þ 417 Inverse Problems in Science and Engineering Proof From (22) and (23), we have eÀ! FðtÞ w b ð gex Þð!, tÞ ¼ c b ð g Þð!, tÞ À w b g !ị g !ịị  ex ! ỵ eÀ!2 FðT Þ FðtÞÀFðT Þ g ð!Þ À gc T0 ẵ lnFT ị=ị FT ị b ex !ịị: Hence, we get w b ð g ÞðÁ, tÞ w b gex ị, tị 2 T0 ẵ lnFT Þ=ފ FðtÞ FðT Þ Downloaded by [Universite Laval] at 01:13 16 July 2014 T0  FtịFT ị FT ị ẵlnFT Þ=ފ g À gex FðtÞÀFðT Þ FðT Þ : ð25Þ From (21) and (22), we obtain w bð!, tÞ b ð gex Þð!, tÞ À w  # " ZT b ! FðT ị !2 Ftị !2 ẵFsịFtịFT ị f!, sị ds ẳ e e gc e ex !ị ỵ FT ị ! asị ! ỵ e t " # Z !2 e!2 FðT Þ T b !2 Ftị !2 ẵFsịFtịFT ị f!, sị c ẳ ds e !ị ỵ e g ex ! þ eÀ! FðT Þ aðsÞ t " # ZT eÀ!2 FðtÞ b !2 FðT Þ !2 Fsị f!, sị c ẳ ds ! e !ị ỵ ! e g ex ! ỵ e!2 FT Þ aðsÞ t " # Z eÀ! FðtÞ t b 2 !2 FðsÞ fð!, sÞ b ¼ ds ! ! e w ð!, 0Þ À : ! ỵ e! FT ị asị Therefore, we have w b ð gex Þð!, tÞ À w bð!, tÞ Zt eÀ!2 FðtÞ b 2 !2 FðsÞ fð!, sÞ b ds ¼ ! ! e w ð!, 0Þ À ! ỵ e!2 FT ị asị 2 eÀ!2 FðtÞ eÀ!2 FðtÞ Z t b !2 Fsị f!, sị b ds 2 ! ỵ ! e w ð!, 0Þ 2 ! þ eÀ! FðT Þ !2 þ eÀ! FðT Þ aðsÞ eÀ!2 FðtÞ eÀ!2 FðtÞ Z T fð!, sÞ !2 Fsị b b!, 0ị ỵ 2T 2 ds ! w ! e ! ỵ e!2 FT ị ! ỵ e!2 FT ị asị # " Z T b 2FðtÞ 2FðtÞÀ2FðT Þ f!, sị 2 !2 w b!, 0ị ỵ T 2T20 FT ị ẵlnFT ị=ị FT ị ds : !2 e! FðsÞ aðsÞ Then, we get w ð gex ÞðÁ, tÞ À wðÁ, tÞ "Z p Ftị FtịFT ị 2T0 FT ị ẵlnFT ị=ị FT ị R 2 ! w b!, 0ị ỵ T ! #12 Z T fð!, sÞ !2 FðsÞ b ds d! ! e aðsÞ 418 P.H Quan et al " #12 Z Z T b pffiffiffi FðtÞ FðtÞÀFðT Þ fð!, sÞ ! Fsị 2T0 FT ị ẵlnFT ị=ị FT ị wxx , 0ị 2 ỵ T ds d! ! e asị R Ftị K1 T0 FT ị ẵlnFT ị=ị FtịFT Þ FðT Þ , ð26Þ where " #12 Z Z T b pffiffiffi fð!, sÞ ! Fsị K1 ẳ wxx , 0ị 2 ỵ T ds d! 1: ! e asị R Thus, from (25) and (26), we obtain w ð g ÞðÁ, tÞ À wðÁ, tÞ w ð g ÞðÁ, tÞ À w ð gex ÞðÁ, tị ỵ w gex ị, tị w, tị 2 FðtÞ Downloaded by [Universite Laval] at 01:13 16 July 2014 T0 ỵ K1 ịFT ị ẵlnFT Þ=ފ FðtÞÀFðT Þ FðT Þ : A numerical experiment Consider the linear homogeneous parabolic equation with time-dependent coefficient uxx x, tị ẳ atịut x, tị, ux, 1ị ẳ gex xị, x, tị R ẵ0, 1, x R, where atị ẳ , 2t ỵ 27ị and x2 ux, 1ị ẳ gex xị ẳ p e 12 : ð28Þ The exact solution of the equation is x2 uex x, tị ẳ p e4t2 þtþ1Þ : 2ðt2 þ t þ 1Þ ð29Þ From (29), we obtain 3!2 b gex !ị ẳ e uex !, 1ị ẳ b : 30ị From (3) and (30), we get Rt aðsÞ ds Rt ðFð1ÞÀFðtÞÞ! 3! b uex !, tị ẳ e e , 31ị where Ftị ẳ ẳ 2s ỵ 1ịds ẳ t ỵ t: Let t ẳ 0, from (31), we have 2 b uex !, 0ị ẳ eF1ịF0ịị! e3! ẳ e! : 32ị Inverse Problems in Science and Engineering 419 Consider the measured data  14 ! g" xị ẳ ỵ " gex xị,  33ị then we have Downloaded by [Universite Laval] at 01:13 16 July 2014 !12  14 Z ỵ1 x 2 pffiffiffi e 12 ds ¼ ": g" À gex ¼ b g" À b gex ẳ "  34ị From (6) and (33), we have the Fourier transform of the regularized solution for the case t ¼  14 ! ub" g" ị!, 0ị ẳ 1ỵ" e3! :  ! ỵ e2!2 Let " be "1 ¼ 10À1, "2 ¼ 10À5, "3 ¼ 10À10, "4 ¼ 10À20, "5 ¼ 10À50, respectively We get the following table for the case t ¼ kb u"i ð gi ÞðÁ, 0Þ À b uex ðÁ, 0Þk2 " À1 "1 ¼ 10 "2 ¼ 10À5 "3 ¼ 10À10 "4 ¼ 10À20 "5 ¼ 10À50 0.038172979714189 1.007122419416841e-005 1.532571311773795e-010 2.262863342628008e-020 1.946443884651801e-032 From Theorem and m ¼ 1/2, we get pffiffiffi Z Z  2 1: Q1=2 ẳ !2 e! F1ịỵ1=2ịb gex !ị d! ẳ !2 e2! d! ẳ R R From (7) and (33), we have the Fourier transform of the regularized solution for the case t ¼  14 ! e À 2! vb" ð g" ị!, 0ị ẳ 1ỵ" : !  "! ỵ e Let " be "1 ¼ 10À1, "2 ¼ 10À5, "3 ¼ 10À10, "4 ¼ 10À20, "5 ¼ 10À50, respectively We get the following table for the case t ¼ " "1 ¼ 10À1 "2 ¼ 10À5 "3 ¼ 10À10 "4 ¼ 10À20 "5 ¼ 10À50 b v"i ð g"i ÞðÁ, 0Þ À b uex ðÁ, 0Þ 2 0.051493210811570 6.033094118141750e-005 7.790717090792106e-009 9.871639018798631e-017 2.672312964517292e-032 420 P.H Quan et al Downloaded by [Universite Laval] at 01:13 16 July 2014 We have the following graph of the Fourier transform of the exact solution b uex ðÁ, tÞ and of the Fourier transform of the regularized solution b u"i g"i ị, tị, i ẳ 1, (Figure 1) We also have the following graph of the Fourier transform of the regularized solution b u"i ð g"i ÞðÁ, tị, i ẳ 3, 4, (Figure 2) Now, Figure can visually represent the Fourier transform of the accuracy solution and the Fourier transform of the regularized solution at initial Figure The Fourier transform of the exact solution b uðx, tÞ and the Fourier transform of the regularized solution ub"i g"i ị:, tị, i ẳ 1, Figure The Fourier transform of the regularized solution ub"i ðg"i Þð:, tÞ, i ¼ 3, 4, Downloaded by [Universite Laval] at 01:13 16 July 2014 Inverse Problems in Science and Engineering 421 Figure The Fourier transform of the exact solution b uðx, 0Þ and the Fourier transform of the regularized solution ub"i g"i ị:, 0ị, i ẳ 1, , Figure The Fourier transform of the exact solution b uðx, tÞ and the Fourier transform of the regularized solution vb"i ðg"i Þð:, tÞ, i ¼ 1, time, t ¼ Note that in Figure 3, the curve number expressing the Fourier transform of the exact solution is indistinguishable from the curve number i expressing the Fourier transform of the regularized solution corresponding to "i, i ¼ 2, , We have the following graph (Figure 4) of the Fourier transform of the exact solution b uex ðÁ, tÞ and of the Fourier transform of the regularized solution b v"i g"i ị, tị, i ẳ 1, 2: Downloaded by [Universite Laval] at 01:13 16 July 2014 422 P.H Quan et al Figure The Fourier transform of the regularized solution vb"i g"i ị:, tị, i ẳ 3, 4, Figure The Fourier transform of the exact solution b uðx, 0Þ and the Fourier transform of the regularized solution ub"i g"i ị:, 0ị, i ẳ 1, , We have the following graph of the Fourier transform of the regularized solution b v"i ð g"i ÞðÁ, tị, i ẳ 3, 4, (Figure 5) Now, the figure can visually represent the Fourier transform of the accuracy solution and the Fourier transform of the regularized solution at initial time, t ¼ (Figure 6) Note that in Figure 6, the curve number expressing the Fourier transform of the exact solution is indistinguishable from the curve number i expressing the Fourier transform of the regularized solution corresponding to "i, i ¼ 2, , Inverse Problems in Science and Engineering 423 Acknowledgement Pham Hoang Quan was supported by the National Foundation for Science and Technology Development (NAFOSTED) Downloaded by [Universite Laval] at 01:13 16 July 2014 References [1] G Clark and C Oppenheimer, 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estimate which is of order T at t 6¼ and an error estimate which is of order ðlnð1ÞÞÀ4 at. .. tị ẳ at ut x, tị, x, tị R  ½0, T Š: 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... 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