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J Inverse Ill-Posed Probl., Ahead of Print DOI 10.1515 / jip-2012-0012 © de Gruyter 2013 On a backward nonlinear parabolic equation with time and space dependent thermal conductivity: Regularization and error estimates Pham Hoang Quan, Dang Duc Trong and Le Minh Triet Abstract We study the inverse problem for nonlinear parabolic equation with variable thermal conductivity a.x; t / which is a ill-posed problem We regularize this problem by using the truncated integral method and then obtain the error estimate between the exact solution and the regularized solution A numerical experiment is given for illustrating our method Keywords Backward problem, ill-posed problem, variable coefficient 2010 Mathematics Subject Classification 35K05, 35K99, 47J06, 47H10 Introduction The backward heat problem (BHP) has been studied by many authors such as Lattes and Lions [3], Showalter [6], Tautenhahn and Schroter [7], Hao [2] in recent years In their researchs, they dealt with BHP by using many methods For example, Showalter used the quasi-reversibility method to consider the final value problem for evolution equations in 1974 (see [6]) In 1996, Tautenhahn and Schroter researched BHP and gave an optimal error estimate for this problem (see [7]) Recently, in 2007, Fu, Xiong and Quian used the Fourier regularization for a backward heat problem in [1] and provided the error estimates between the exact solution and the approximate solution However, they dealt with the problem for the parabolic equations with constant coefficient Hence, we propose to examine the general parabolic equation with non-constant coefficient In fact, there are some papers related to BHP with non-constant coefficient Recently, in [5], we have considered the inverse problem for a parabolic equation with time-dependent coefficient a.t /u t x; t/ D uxx x; t /; x; t/ R u.x; T / D g.x/; Œ0; T /; x R; All authors were supported by the National Foundation for Science and Technology Development (NAFOSTED) Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM P H Quan, D D Trong and L M Triet and obtained error estimates of Hölder type and logarithm type between the exact solution and the regularized solution In [2], Hao and Duc dealt with a regularization method for an ill-posed backward parabolic equation with time-dependent coefficient ´ u t C A.t/u D 0; < t < T; (1.1) ku.T / f kH Ä ; f H; where H is a Hilbert space and A.t / (0 < t < T ) is a positive self-adjoint unbounded operator from D.A.t // H to H and f is a given data In [2], the authors considered the well-posed problem (see [2, page 8]) ´ w t C B.t /w D 0; < t < T; w.T / D f; ˛ > 0; in which ´ B.t / D A.t /; A.2T Ä t Ä T; t /; T < t Ä 2T: Then they let w.2T / D g and suggested the regularized problem as follows: ´ v t C B.t /v D 0; < t < T; (1.2) ˛v.0/ C v.2T / D g; ˛ > 0: In [2], they proved that problem (1.2) is well-posed and gave Hölder type of error estimate between the regularized solution v and the exact solution u (see [2, Theorem 3.4]) under some assumptions (see [2, conditions 3.1, 3.2, page 7]) of the operator A We can see that Hao and Duc have considered the problem when a.x; t/ is not bounded Up to now, there are many papers related to the inverse problem for parabolic equation with constant coefficient (see [4, 7–9]) On the other hand, there are few papers related to the case of time-dependent coefficient (such as [2, 5], etc.) Thus, we wish to investigate the inverse problem for the parabolic problem u t x; t/ a.x; t /uxx x; t / D f x; t; u; ux ; uxx /; u.x; T / D g.x/; x; t/ R Œ0; T /; (1.3) x R; (1.4) where f x; t; u; ux ; uxx / and a.x; t / are given such that there exist p; q; L > satisfying < p Ä a.x; t/ Ä q (1.5) and jf x; t; u1 ; v1 ; w1 / f x; t; u2 ; v2 ; w2 /j Ä L.ju1 v2 j C jv1 v2 j C jw1 w2 j/; Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM On a BNPE with time and space dependent thermal conductivity for all x; t; u1 ; v1 ; w1 /; x; t; u2 ; v2 ; w2 / R Œ0; T R3 Noting that we consider the solution in H R/ and the data in L2 R/ We can see that heat transfer coefficient of (1.3) is a function which depends on space and time It is a new point in this paper Throughout this paper, we define the Fourier transform F W L2 R/ ! L2 R/ as follows: Z C1 F f / / D p f x/e i x d x: In this paper, we assume that k.t / D limx!1 a.x; t/ and let b.x; t / D a.x; t / k.t/: From (1.5), we can see that < p Ä k.t / Ä q It follows that jb.x; s/j D ja.x; s/ for all x; s/ R Then, we have u t x; t/ k.s/j Ä ja.x; s/j C jk.s/j Ä 2q; (1.6) Œ0; T : k.t/uxx x; t / D '.u; ux ; uxx /.x; t/; u.x; T / D g.x/; x; t/ R Œ0; T /; x R; (1.7) (1.8) in which '.u; ux ; uxx /.x; t / D b.x; t /uxx x; t/ C f x; t; u; ux ; uxx /: Using the Fourier transform, we can clearly find out the form of the solution of (1.3)–(1.4) u.x; t / D P x; t / K.x; t; u/; (1.9) where P x; t/ D p Z K.x; t; u/ D p Z C1 Á.T / e Á.t// F g/ /e i x d ; (1.10) C1 Z T e Á.s/ Á.t// t F '.u; ux ; uxx // ; s/dse i x d (1.11) and Z Á.t / D t k.s/ds: (1.12) In this paper, we shall use the truncated integral method for regularizing problem (1.9) Then, we suggest the regularized problem for (1.9) as u x; t / D P x; t / K x; t; u /; (1.13) Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM P H Quan, D D Trong and L M Triet in which P x; t/ D p Z K x; t; u / D p Z C1 Á.T / e Á.t// F g/ / Œ a ;a /e i x d ; (1.14) C1 Z T e Á.s/ Á.t// F '.u ; u x ; u xx // ; s/ds t Œ a ;a /e i x d ; (1.15) and a shall be chosen later such that a ! as ! 0: The remainder of this paper is divided into two sections In Section 2, we shall give main results of regularizing problem (1.3)–(1.4) and the proof of main results Then we consider problem (1.7)–(1.8) for the case that ' D and a.t/ is perturbed A numerical test is given in Section in order to illustrate our method In fact, Sections and contain new ideas which are the method of transforming problem (1.3)–(1.4) into problem (1.7)–(1.8) (in Section 1) and applying the norm k kH R/ in the space H R/ (in all of theorems in Section 2) Regularization and error estimates Throughout this paper, we let ; a be fixed positive numbers, T1 D max¹Á.T /; 1º and R.x/ D C x C x For short notation, we define '.u ; u x ; u xx / ; s/ D '.u / ; s/ and the norm in H R/ as follows: 2 kf kH R/ D kf kL R/ C kfx kL2 R/ C kfxx kL2 R/ Á 12 : Next, we get the existence and the uniqueness of the solution of problem (1.13) in the following theorem Theorem 2.1 Let g L2 R/ and a ; / be a function satisfying (1.5) Then problem (1.13) has a unique solution u C.Œ0; T I H R// Proof Put W u/.x; t / D P x; t / K x; t; u/; where P x; t/ D p Z C1 e Á.T / Á.t// F g/ / Œ a ;a /e i x d Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM On a BNPE with time and space dependent thermal conductivity and K x; t; u/ D p Z C1 ÄZ T e Á.s/ Á.t// F '.u// ; s/ds t Œ a ;a We claim that for every u; v C.Œ0; T I H R//, k kW k u/ ; t / Ä /e i x d : 1, we have W k v/ ; t /kH R/ t/k T kŠ T k K 2k Rk a /e 2ka Á.T / (2.1) jku vjk2 ; p where K D 3.L C 2q/ and jk jk is supremum norm in C.Œ0; T I H R//: We prove (2.1) by induction principle For k D 1, we have kW u/ ; t / W v/ ; t /kH R/ Z D R / jF W u// ; t/ F W v// ; t/j2 d : Thus, we obtain kW u/ ; t/ W v/ ; t /kH R/ Z D R / Œ a ;a / Ä T ˇ2 ˇ F '.v// ; s/jds ˇˇ d ˇZ T ˇ ˇ e Á.s/ Á.t// jF '.u// ; s/ ˇ t Z t/ R / Œ a ;a / T ÄZ e2 Á.s/ Á.t// t jF '.u// ; s/ F '.v// ; s/j2 ds d : jF '.u// ; s/ F '.v// ; s/j2 d Hence, we get W v/ ; t/kH R/ Z T ÄZ Ä e 2a Á.T / T t/R.a / kW u/ ; t/ D e 2a Á.T / T t Z T ÄZ t /R.a / t j'.u/.x; s/ ds '.v/.x; s/j2 dx ds: Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM P H Quan, D D Trong and L M Triet Therefore, we have W v/ ; t /kH R/ kW u/ ; t / Äe 2a2 Á.T / T T Z t /K R.a / t Ä e 2a Á.T / t /K R.a /jku T T ku ; s/ v ; s/kH R/ ds vjk2 : Thus, (2.1) holds for k D Suppose that (2.1) holds for k D n, we prove that (2.1) holds for k D n C In fact, we get W nC1 u/ ; t/ W nC1 v/ ; t / Z D R / Œ a ;a / H R/ ˇZ ˇ ˇ ˇ T Á.s/ e Á.t// t ˇ2 ˇ jF '.W u/// ; s/ F '.W v/// ; s/jds ˇˇ d : n n Then we have kW nC1 u/ ; t/ W nC1 v/ ; t /kH R/ Z Ä T t/ R / Œ a ;a / T ÄZ Á.s/ e2 t Ä T jF '.W n u/// ; s/ F '.W n v/// ; s/j2 ds d t/R.a /e 2a Á.T / Z ÄZ T jF '.W n u/// ; s/ D T Á.t// F '.W n v/// ; s/j2 ds d t t/R.a /e 2a Á.T / Z T ÄZ jF '.W n u/// ; s/ t F '.W n v/// ; s/j2 d ds: Hence, we obtain kW nC1 u/ ; t/ Ä T W nC1 v/ ; t /kH R/ t/R.a /e 2a Á.T / Z T ÄZ j'.W n u//.x; s/ '.W n v//.x; s/j2 dx ds t Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM On a BNPE with time and space dependent thermal conductivity Ä K T t/R.a /e 2a Á.T / T Z t kW n u/ ; s/ Ä T nC1 K 2nC2 RnC1 a /e 2.nC1/a D Á.T / jku vjk2 W n v/ ; s/kH R/ ds T Z s/n T t T t/nC1 nC1 2nC2 nC1 T K R a /e 2.nC1/a Á.T / jku n C 1/Š nŠ ds vjk2 : Therefore, by the induction principle, we have kW k u/ ; t/ Ä T W k v/ ; t/kH R/ t /k=2 k=2 k k=2 T K R a /e ka Á.T / jku p kŠ vjk; for all u; v C.Œ0; T I H R//: Then we obtain jkW k u/ Tk W k v/jk Ä p K k Rk=2 a /e ka Á.T / jku kŠ vjk: We consider W W C.Œ0; T I H R// ! C.Œ0; T I H R//: From Tk p K k Rk=2 a /e ka Á.T / ! 0; kŠ when k ! 1, there exists a positive integer number k0 such that T k0 p K k0 Rk0 =2 a /e k0 a Á.T / < 1; k0 Š and W k0 is a contraction We include that the equation W u / D u has a unique solution u C.Œ0; T I H R//: The proof of Theorem 2.1 is completed In next theorem, we shall prove the stability of the modified method for problem (1.13) Theorem 2.2 Let a ; / be as in Theorem 2.1, be a positive number, g and g be in L2 R/ such that kg g kL2 R/ Ä Suppose that u and v are defined by (1.13) corresponding to final values g and g in L2 R/, respectively Then we obtain p p 2 ku ; t/ v ; t/kH R/ Ä 2e a Á.T / Á.t// R.a /e K T R.a / ; (2.2) in which a is chosen later such that a ! as ! 0: Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM P H Quan, D D Trong and L M Triet Proof From (1.13), we construct the regularized solution corresponding to the exact data Z a Ä u x; t/ D p e Á.T / Á.t// F g/ / a (2.3) Z T Á.s/ Á.t// i x e F '.u // ; s/ds e d ; t and the regularized solution corresponding to the measured data Z a Ä e Á.T / Á.t// F g / / v x; t/ D p a (2.4) Z T Á.s/ Á.t// i x e F '.v // ; s/ds e d : t From (2.3) and (2.4), we get the following estimate: ku ; t/ v ; t /kH R/ Z C1 D R / jF v / ; t / a Z Ä2 ˇ ˇ R /ˇe a Z C2 a a Á.T / ˇZ ˇ R /ˇˇ F u / ; t/j2 d Á.t// T e F g / / Á.s/ Á.t// ˇ2 ˇ F g/ //ˇ d F '.v // t ˇ2 ˇ F '.u /// ; s/ds ˇˇ d : Therefore, we have ku ; t/ Ä 2e v ; t /kH R/ 2a2 Á.T / Á.t// Z a j.F g / / R.a / a C 2.T F g/ //j2 d t /R.a /e 2a Á.t/ Z T Z a e 2a Á.s/ j.F '.v // t F '.u /// ; s/j2 d ds a D N1 t/ C N2 t/ where N1 t/ D 2e 2a Á.T / Á.t// Z a R.a / a jF g / / F g/ /j2 d (2.5) Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM On a BNPE with time and space dependent thermal conductivity and t /R.a /e 2a Á.t/ Z T Z a e 2a Á.s/ j.F '.v // N2 t/ D 2.T t (2.6) F '.u /// ; s/j2 d ds: a From (2.5), we get N1 t/ Ä 2e 2a Á.T / Ä 2e 2a Á.T / Á.t// C1 Z jF g / / R.a / Á.t// F g/ /j2 d gkL R/ : R.a /kg (2.7) and (2.6) implies N2 t/ Ä 2.T 2a2 Á.t/ t/R.a /e T Z e 2a Á.s/ t C1 Z Ä 2K T Z j.F '.v // t /R.a /e T e 2a Á.s/ t F '.u /// ; s/j2 d ds 2a2 Á.t/ ku ; s/ v ; s/kH R/ d ds (2.8) From (2.7) and (2.8), we have e 2a Á.t / ku ; t/ Ä 2e 2a Á.T / v ; t /kH R/ gkL R/ Z T C 2K TR.a / e 2a Á.s/ ku ; s/ R.a /kg t v ; s/kH R/ ds: Using Gronwall’s inequality, we get the estimate ku ; t/ 2a v ; t/kH R/ Ä 2e Á.T / Á.t// R.a /e 2K T R.a / : Hence, we obtain ku ; t/ v ; t /kH R/ Ä p 2e a Á.T / Á.t// p R.a /e K T R.a / : The proof of Theorem 2.2 is completed Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM 10 P H Quan, D D Trong and L M Triet Then we give the estimate between the regularized solution corresponding to the exact data and the exact solution as follows Theorem 2.3 Suppose that a ; / is a function satisfying (1.5) and ˛; m are positive numbers Let u be defined by (1.13) corresponding to the final value g in L2 R/ and u be the exact solution of (1.3)–(1.4) satisfying ³ ²Z 4C˛ e 2m j j jF u/ ; t/j2 d W t Œ0; T < 1: C˛;m0 D sup R in which m D Á.T / C m C (i) If < > 2 4C˛ 3KmT / ˛ < min¹e ;e º, then we obtain q u ; t /kH R/ Ä C˛;m0 ku ; t/ m ; (2.9) for all t Œ0; T /: (ii) If < 3KmT / < min¹e ku ; t/ 4C˛ ˛ ; e e º, then we obtain u ; t /kH R/ Ä q  C˛;m ÃÁ.t/C m2 ln / ; (2.10) for all t Œ0; T /: Proof (i) From (1.9) and (1.13), we obtain ku ; t/ u ; t/kH R/ Z C1 D C C /jF u / ; t / F u/ ; t/j2 d Z D C C / jF u/ ; t /j2 d RnŒ a ;a Z a C a C ˇZ ˇ ˇ ˇ T e C / Á.s/ Á.t// F '.u// ; s/ t ˇ2 ˇ F '.u // ; s//ds ˇˇ d D I1 t/ C I2 t/ in which (2.11) Z I1 t / D RnŒ a ;a C C /jF u/ ; t/j2 d (2.12) Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM 13 On a BNPE with time and space dependent thermal conductivity Because < < min¹e ku ; t/ 3KmT / 4C˛ ˛ ; e º, we have q 4C˛ 2 Ä C˛;m0 e a Á.t/Cm/ e K T a q 4C˛ m 4C˛ Ä C˛;m0 e a Á.t/Cm/ e a q 4C˛ m D C˛;m0 e Á.t/C /a : u ; t /kH R/ (2.17) Let a D ln // 4C˛ and (2.17), we obtain ku ; t / u ; t /kH R/ Ä q C˛;m0 Á.t/C m : (ii) Let a D ln.ln /// 4C˛ and (2.17), we get ku ; t / u ; t/kH R/ Ä q  C˛;m0 ÃÁ.t/C m2 : ln / The proof of Theorem 2.3 is completed Finally, we prove the error estimate between the regularized solution corresponding to the measured data g and the exact solution of problem (1.3)–(1.4) Theorem 2.4 Suppose that a ; / is a function satisfying (1.5), ˛, m are positive numbers, g; g are in L2 R/ such that kg g kL2 R/ Ä and u , v are defined by (1.13) corresponding to final values g; g in L2 R/ respectively Let u be the exact solution of (1.3)–(1.4) such that ²Z ³ 4C˛ 2m j j C˛;m0 D sup e jF u/ ; t/j d W t Œ0; T < R in which m D Á.T / C m C > 0: (i) If 0< ² < e 3K T min¹ ˇ/;mº 4C˛ ˛ ;e 3T1 ˇ 4C˛ 2C˛ ;e ³ where ˇ 0; 1/, then we have ku ; t/ u ; t /kH R/ Ä in which m1 t / D min¹ˇ; Á.t / C m1 t/  p 4C˛ Á.t/ 6e Œln / C q à C˛;m ; m º: Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM 14 P H Quan, D D Trong and L M Triet (ii) If 3K T min¹ ˇ/;mº ² < e 0< 4C˛ ˛ 3T1 ˇ ;e 4C˛ 2C˛ ;e e ³ where ˇ 0; 1/, then we have ku ; t/ u ; t/kH R/ Ä p 6e ln.ln /// 4C˛ Á.t/ ˇ C in which m1 t / D min¹ˇ; Á.t / C q  C˛;m0 ÃÁ.t/C m2 ln / ; m º: Proof (i) From Theorem 2.2 and Theorem 2.3 (i), we get the following estimate: kv ; t/ u ; t /kH R/ Ä kv ; t / u ; t /kH R/ C ku ; t/ u ; t/kH R/ q p 2 m Ä 6e a Á.T / Á.t// a2 e K T a C C˛;m0 Á.t/C q p 4C˛ m C C˛;m0 Á.t/C : Ä 6e a Á.t/ e ˇ /a (2.18) Put a D ln // 4C˛ ; we obtain kv ; t/ p q 4C˛ m Á.t/ ˇ 6e Œln / C C˛;m0 Á.t/C  à q p 4C˛ Á.t/ Ä m1 t/ 6e Œln / C C˛;m0 ; u ; t/kH R/ Ä where m1 t/ D min¹ˇ; Á.t / C (ii) Let m º:    ÃÃà 4C˛ a D ln ln : By (2.18), we get kv ; t/ Ä Ä u ; t /kH R/ p p Ä Â Ã 1 4C˛ Á.t/ 6e Œln.ln // ln 4C˛ Á.t/ 6e Œln.ln // ˇ C q ˇ Ä q C C˛;m0 ln / Ä C˛;m0 ln / Á.t/C m Á.t/C m : The proof of Theorem 2.4 is completed Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM On a BNPE with time and space dependent thermal conductivity 15 Remark 2.5 (i) In Theorem 2.4, we require a strong condition on the exact solution of problem (1.3)–(1.4) Thus, it is a weak point of this paper (ii) In this paper, we want to focus on physical meaning of variable coefficient In fact, heat distribution in a body depends on many elements (e.g heat source, time, space, etc.) One important element is a coefficient of heat conductivity, but it is often not a constant because a body is usually nonhomogeneous Moreover, we note that equation (1.3) contains the backward parabolic equation (f D 0) and the forward parabolic equation (f D 2a.x; t /uxx x; t/) (iii) On the other hand, we are interested in developing problem in [5] with time and space dependent coefficient and nonlinear heat source By using F x; t; u; ux ; uxx / D f x; t; u; ux ; uxx / C a.x; t/ 1/uxx x; t/; we can transform problem (1.3)–(1.4) into the case constant coefficient while the function F still satisfies the same Lipschitz condition as the function f However, we would like to develop problem (1.7)–(1.8) when a coefficient of heat conductivity a.x; t/ can be perturbed (see Theorem 2.7 in this paper) Therefore, we transform problem (1.3)–(1.4) into problem (1.7)–(1.8) Next, we give one example for the case that a coefficient of heat conductivity a.t/ is perturbed (problem (2.19)– (2.20)) In future, we shall develop problem (2.19)–(2.20) for the case a.x; t/ is perturbed In fact, if we consider the problem a.t/rxx x; t / r t x; t / D 0; x; t/ R r.x; T / D g.x/; Œ0; T ; x R; (2.19) (2.20) where a.t/ > is a continuous function on Œ0; T and g L2 R/: Noting that g; a/ is given data in problem (2.19)–(2.20) We assume that u is the exact solution of problem (1.3)–(1.4) corresponding to the exact data g; a and g ; a are measured data satisfying kg g kL2 R/ Ä and ka a kC Œ0;T Ä Using Fourier transform, we obtain the solution of problem (2.19)–(2.20) Z C1 r.x; t/ D p e Á.T / Á.t// F g/ /e i x d ; (2.21) Rt where Á.t/ D a.s/ds: Then we construct the regularized solution corresponding to measured data g ; a by applying the truncated method Z C1 r g ; a /.x; t/ D p e Á T / Á t// F g / / (2.22) ei x Œ b ;b /d ; Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM 16 P H Quan, D D Trong and L M Triet where Á t/ D Rt a s/ds: Applying triangle inequality, we have kr g ; a / ; t / r ; t /kL2 R/ D kF r /.g ; a / ; t / F r/ ; t/kL2 R/ Ä kF r /.g ; a / ; t / F r /.g; a / ; t/kL2 R/ C kF r /.g; a / ; t/ C kF r /.g; a/ ; t / F r /.g; a/ ; t/kL2 R/ F r/ ; t/kL2 R/ : (2.23) We get the following lemma for regularizing problem (2.19)–(2.20) Lemma 2.6 Let > 0, b > satisfy lim !0 b D C1 and lim !0 b D If we assume that a ; a are the measured data and the exact data of problem (2.19)– (2.20) respectively, then there exists ı1 > such that for every 0; ı1 / ˇ R s a !/ a.!//d! ˇ ˇe t 1ˇ Ä 2T b for all Ä t Ä s Ä T; Œ b ; b : Proof First, we have ja !/ a.!/j Ä ka akC Œ0;T Ä : It implies that s Z T Ä T t/ Ä t/ Ä s a !/ t a.!//d! Ä s t/ Ä T: Then we get e T b2 1Äe It follows that ˇ R s a !/ ˇe t T 1Äe a.!//d! Rs t a !/ a.!//d! ˇ ˇ 1ˇ Ä h t / D max¹ˇe By using the limits ˇ b2 T ˇ ˇe 1ˇ lim ˇ ˇ D and ˇ b Tˇ !0 1Äe b2 T T 1Äe ˇ ˇ 1ˇ; ˇe b T T b2 1: ˇ 1ˇº: ˇ b2 T ˇ ˇe 1ˇ lim ˇ ˇ D 1; ˇb T ˇ !0 it implies that there exists a ı1 > such that for all 0; ı1 / 8ˇ ˇ ˇe b T 1ˇ ˆ ˆ ˆ ˆ < j b T j Ä 2; ˇ b2 T ˇ ˆ ˇe ˇ ˆ ˆ ˆ Ä 2: : jb T j Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM On a BNPE with time and space dependent thermal conductivity 17 It implies that 8ˇ < ˇe b : ˇˇe b Then we get ˇ R s a ˇe t T ˇ 1ˇ Ä 2b T; ˇ 1ˇ Ä 2b T: T !/ a.!//d! ˇ 1ˇ Ä h t/ Ä 2b T: The proof of Lemma 2.6 is completed Theorem 2.7 Let > 0, t 0; T /, M D max t 2Œ0;T ja.t/j, g ; g L2 R/ and a; a C Œ0; T satisfy kg gkL2 R/ Ä and ka akC Œ0;T Ä respectively If we assume that the exact solution r ; t/ of problem (2.19)–(2.20) satisfies C1 t/ D C 1 kr ; T /kL2 R/ C 2M T / kr ; t/kH R/ M C kr ; 0/kL2 R/ < 1; then there exists a ı > such that for every kr g ; a / ; t / Proof Let < 0; ı/ r ; t /kL2 R/ Ä C1 t/ ln / : Ä max t 2Œ0;T a.t / For all t Œ0; T we get ja t /j ja.t /j Ä ka akC Œ0;T < Ä max a.t/: t 2Œ0;T It follows that ja t /j < max a.t / C max a.t/ D M: t 2Œ0;T t 2Œ0;T From (2.22), we construct the regularized solution corresponding to the data g ; a /, g; a / and g; a/ respectively as follows: Z C1 e Á T / Á t// F g / / r g ; a /.x; t / D p (2.24) e i x Œ b ;b /d ; Z C1 r g; a /.x; t / D p e Á T / Á t// F g/ / (2.25) i x e Œ b ;b /d ; Z C1 r g; a/.x; t / D p e Á.T / Á.t// F g/ / (2.26) e i x Œ b ;b /d : Rt where Á t/ D a s/ds Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM 18 P H Quan, D D Trong and L M Triet From (2.24) and (2.25), we get F r g; a // ; t/j2 jF r g ; a // ; t/ D e2 Á T / Á t// D e2 R T a s/ds Ä e2 M.T t t/ F g/ /j2 jF g / / F g/ /j2 jF g / / jF g / / F g/ /j2 Œ b ;b Œ b ;b Œ b ;b / / /: Therefore, we have kF r g ; a // ; t / F r g; a // ; t/kL R/ Z b Ä e M.T t/ jF g / / F g/ /j2 d b Äe 2b M.T t/ Ä e 2b 2M T : Then we obtain F r /.g; a / ; t/kL2 R/ Ä e b kF r /.g ; a / ; t / 2M T : (2.27) From (2.25), (2.26) and Lemma 2.6, there exists a ı1 > such that for all 0; ı1 / jF r g; a // ; t / F r g; a// ; t/j ˇ ˇ D ˇe Á T / Á t// e Á.T / Á.t// ˇjF g/ /j Œ b ;b / ˇ ˇ D e Á.T / Á.t// ˇe Á T / Á.T / Á t/CÁ.t// 1ˇjF g/ /j Œ b R ˇ ˇ 2RT T D e t a.s/ds ˇe t a s/ a.s//ds 1ˇjF g/ /j Œ b ;b / Ä eb TM 2b T jF g/ /j Œ b ;b ;b / /: Then we get kF r g; a // ; t/ F r g; a// ; t/kL2 R/ ÄZ b Á2 Ä e b TM :2b T jF g/ /j d b Ä 2b TM D 4b e D 2b e b T 2 Z b b TM 2 jF g/ /j d T kr ; T /kL2 R/ : (2.28) Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM On a BNPE with time and space dependent thermal conductivity 19 From (2.21) and (2.26), we have kF r /.g; a/ ; t / F r/ ; t /kL R/ Z Ä e Á.T / Á.t// jF g/ /j2 Œ Z C 2b Á.t/ Z Œ C b b ;b RnŒ Äe Œ b ;b /d b ;b jF r/ ; t/j2 d Á.T / e2 b ;b Z RnŒ b ;b jF g/ /j2 d j F r/ ; t/j2 d : It implies that F r/ ; t/kL R/ kF r /.g; a/ ; t/ Äe 2b Á.t/ De 2b Á.t/ kF r/ ; 0/kL R/ C kr ; 0/kL R/ C kF r/ ; t/kH R/ b2 kr ; t/kH R/ : b2 (2.29) From (2.23), (2.27), (2.28) and (2.29), we obtain the following estimate: kr g ; a / ; t / Äe r ; t /kL2 R/ b2 M T C 2b e b C If we choose b D 1n 2M T kr g ; a / ; t / Ä p p C 2M T T kr ; T /kL2 R/ kr ; t /kH R/ b Ce b Á.t/ kr ; 0/kL2 R/ : , then we get r ; t /kL2 R/ ln / kr ; T /kL2 R/ M C 2M T / kr ; t/kH R/ ln / C Á.t/ MT kr ; 0/kL2 R/ : Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM 20 P H Quan, D D Trong and L M Triet Moreover, for every t 0; T / we have  Ã1 ˆ p ˆ ˆ ˆ ln D 0; lim ˆ ˆ !0 ˆ ˆ ˆ ˆ  Ã3 < p lim ln D 0; ˆ !0 ˆ ˆ ˆ ˆ ˆ  Ã1 ˆ ˆ Á.t/ ˆ ˆ M T ln D 0: : lim !0 It implies that there exists a ı2 > such that for every p ˆ ˆ ; < ˆ ˆ ˆ ln / ˆ ˆ ˆ ˆ < p  1à ln ; < 12 ˆ ln / ˆ ˆ ˆ ˆ ˆ Á.t / ˆ ˆ MT < ˆ : : ln / Let ı D min¹ı1 ; ı2 ; M º; then for every kr g ; a / ; t / 0; ı2 /, 0; ı/, we have r ; t /kL2 R/ Ä C1 t/ ln / where 1 kr ; T /kL2 R/ C 2M T / kr ; t/kH R/ C kr ; 0/kL2 R/ : M The proof of Theorem 2.7 is completed C1 t/ D C Numerical experiment Consider the nonlinear parabolic problem with time dependent coefficient u t x; t/ 2t C 1/uxx x; t / D f x; t; u; ux ; uxx /; x R Œ0; T ; Z 1 u.x; T / D p e i x d ; x R; 6/ exp (3.1) (3.2) where f x; t; u; ux ; uxx / D p Z 1Ä 1 exp / 2t C 1/F uxx / / e i x d : Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM On a BNPE with time and space dependent thermal conductivity 21 The Fourier transform of the exact solution of problem (3.1)–(3.2) is F u/.x; t / D t C1 : exp / (3.3) Let t D 0; from (3.3), we have F u/.x; 0/D exp /: From (3.2), we consider the measured data à  u.x; T /: u x; T / D C kgkL2 R/ (3.4) Then we have ku ; T / u ; T /kL2 R/ D : From (1.13) and (3.4), we have the regularized solution corresponding to the measured data v x; T / Z a v x; t / D p W v / ; t/e i x d ; (3.5) a where W v / ; t / D e Á.T / Á.t// F u ; T // / T Z e Á.s/ Á.t// F '.v // ; s/ds: t From Theorem 2.1, we use the iteration for (3.5) as follows: v ;0 x; t / D 0; v ;m x; t / D p Z a W v ;m / ; t/e i x d : a We consider D 10 ; D 10 ; D 10 ; D 10 ; D 10 and m D Then we propose the error estimate between the exact solution and the regularized solutions corresponding to i , i D 1; : : : ; From (3.3), (3.5) and   Ãà 4C˛ a D ln ; we get the following table which expresses the error estimate for the case t D and t D 0:5 Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM 22 P H Quan, D D Trong and L M Triet a 10 10 10 10 10 kv i ;m 1.1815 1.3572 1.4719 1.5590 1.6302 ; 0/ u ; 0/kH R/ kv i ;m 2.359138e-001 1.350869e-001 1.263248e-001 1.254694e-001 1.253841e-001 ; 0:5/ u ; 0:5/kH R/ 2.132146e-001 1.056336e-001 9.701422e-002 9.618007e-002 9.609694e-002 Then we plot the graph of the Fourier transforms of the exact solution and the regularized solution v i ;m , m D at the time t D in order to illustrate our method visually (see Figure 1) Next, the graph of the Fourier transforms of the exact solution and the regularized solution v i ;m ; m D at the time t D 0:5 is given (see Figure 2) In this paper, we use the truncated integral method for approximating the exact solution Hence, the rate of convergence depends on choosing a If we choose another form of a , we shall give another rate of convergence From (3.3), (3.5) and    ÃÃà 4C˛ a D ln ln ; we get the following table which expresses the error estimate for the case t D and t D 0:5 a 10 10 10 10 10 0.9644 1.0884 1.1409 1.1730 1.1956 kv i ;m ; 0/ u ; 0/kH R/ kv i ;m 3.664731e-001 1.780101e-001 1.412091e-001 1.332348e-001 1.282019e-001 ; 0:5/ u ; 0:5/kH R/ 4.860941e-001 2.126704e-001 1.430881e-001 1.206746e-001 1.093604e-001 Next, we also plot the graph of the Fourier transforms of the exact solution and the regularized solution v i ;m , m D at the time t D for the case    ÃÃà 4C˛ a D ln ln (see Figure 3) Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM On a BNPE with time and space dependent thermal conductivity 23 Figure The Fourier transform of the exact solution and the regularized solution v i ;m , m D at the time t D Figure The Fourier transform of the exact solution and the regularized solution v i ;m , m D at the time t D 0:5 Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM 24 P H Quan, D D Trong and L M Triet Finally, the graph of the Fourier transforms of the exact solution and the regularized solution v i ;m , m D at the time t D 0:5 for the case    ÃÃà 4C˛ a D ln ln is given (see Figure 4) Figure The Fourier transform of the exact solution and the regularized solution v i ;m , m D at the time t D for the case a D ln.ln /// 4C˛ Figure The Fourier transform of the exact solution and the regularized solution v i ;m ; m D at the time t D 0:5 for the case a D ln.ln /// 4C˛ Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM On a BNPE with time and space dependent thermal conductivity 25 Remark From (1.9) and (3.4), we obtain the exact solution corresponding to the measured data u x; T / Z 1 w x; t/ D p W w / ; t/e i x d ; where W w / ; t/ D e Á.T / Á.t// F u ; T // / T Z e Á.s/ Á.t// (3.6) F '.w // ; s/ds: t Now, we cannot calculate (3.6) exactly (we need to find F '.w // ; s/ while we have not known w yet) From Theorem 2.1, we use the iteration for (3.5) as follows: w ;0 x; t/ D 0; w ;m x; t/ D p Z a W w ;m / ; t/e i x d : a Then we calculate the error between the exact solution corresponding to g and the exact solution corresponding to g in the following table kwıi ;5 ; 0/ ı 10 10 10 10 10 u ; 0/kH R/ 3.178231e+000 2.795433e+000 2.758505e+000 2.754826e+000 2.754458e+000 In the following table, we can show the rate kwıi ;4 ; 0/ u ; 0/kH R/ kvıi ;4 ; 0/ u ; 0/kH R/ in order to compare two errors We can see that the error kwıi ;4 ; 0/ u ; 0/kH R/ is larger than the error kvıi ;4 ; 0/ u ; 0/kH R/ many times Therefore, problem (2.19)–(2.20) is illposed Noting that we theoretically consider the problem for parabolic equation Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM 26 P H Quan, D D Trong and L M Triet kwıi ;5 ;0/ u ;0/k H R/ ı 10 10 10 10 10 kvıi ;5 ;0/ u ;0/k H R/ 3:178231eC000 2:132146e 001 2:795433eC000 1:056336e 001 2:758505eC000 9:701422e 002 2:754826eC000 9:618007e 002 2:754458eC000 9:609694e 002 Ð 14:906 Ð 26:463 Ð 28:434 Ð 28:642 Ð 28:663 with time and space dependent coefficient However, we only investigate the nonlinear parabolic equation with time-dependent coefficient It is a weak point of this numerical test in this paper Bibliography [1] C.-L Fu, X.-T Xiong and Z Qian, Fourier regularization for a backward heat equation, Journal of Mathematical Analysis and Applications 331 (2007), no 1, 472–480 [2] D N Hao and N V Duc, Stability results for backward parabolic equations with time-dependent coefficients, Inverse Problems 27 (2011), no 2, Article ID 025003 [3] R Lattes and J L Lions, Méthode de quasi-reversibilite et applications, Dunod, Paris, 1967 [4] P T Nam, D D Trong and N H Tuan, The truncation method for a two-dimensional nonhomogeneous backward heat problem, Applied Mathematics and Computation 216 (2010), 3423–3432 [5] P H Quan, D D Trong, L M Triet, N H Tuan, A modified quasi-boundary value method for regularizing of a backward problem with time-dependent coefficient, Inverse Problems in Science and Engineering 19 (2011), no 3, 409–423 [6] R E Showalter, The final value problem for evolution equations, Journal of Mathematical Analysis and Applications 47 (1974), 563–572 [7] U Tautenhahn and T Schroter, On optimal regularization methods for the backward heat equation, Zeitschrift für Analysis und ihre Anwendungen 15 (1996), 475–493 [8] D D Trong, P H Quan, T V Khanh and N H Tuan, A nonlinear case of the 1-D backward heat problem: Regularization and error estimate, Zeitschrift für Analysis und ihre Anwendungen 26 (2007), no 2, 231–245 [9] D D Trong, P H Quan and N H Tuan, A final value problem for heat equation: regularization by truncation method and new error estimates, Acta Universitatis Apulensis 22 (2010), 41–52 Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM On a BNPE with time and space dependent thermal conductivity 27 Received February 28, 2012 Author information Pham Hoang Quan, Department of Mathematics and Applications, SaiGon University, 273 An Duong Vuong, Ho Chi Minh City, Vietnam E-mail: quan.ph@cb.sgu.edu.vn Dang Duc Trong, Department of Mathematics, University of Science, Vietnam National University, 227 Nguyen Van Cu, Dist 5, Ho Chi Minh City, Vietnam Le Minh Triet, Department of Mathematics and Applications, SaiGon University, 273 An Duong Vuong, Ho Chi Minh City, Vietnam E-mail: triet.lm@cb.sgu.edu.vn Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 11/18/13 10:13 AM ... Bibliography [1] C.-L Fu, X.-T Xiong and Z Qian, Fourier regularization for a backward heat equation, Journal of Mathematical Analysis and Applications 331 (2007), no 1, 472–480 [2] D N Hao and N... Date | 11/18/13 10:13 AM On a BNPE with time and space dependent thermal conductivity 27 Received February 28, 2012 Author information Pham Hoang Quan, Department of Mathematics and Applications,... E Showalter, The final value problem for evolution equations, Journal of Mathematical Analysis and Applications 47 (1974), 563–572 [7] U Tautenhahn and T Schroter, On optimal regularization methods