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

Applied Mathematics and Computation 219 (2013) 6066–6073 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc On a backward heat problem with time-dependent coefficient: Regularization and error estimates Nguyen Huy Tuan a, Pham Hoang Quan b, Dang Duc Trong c, Le Minh Triet b,⇑ 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 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 b a r t i c l e i n f o 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 Keywords: Backward problem Fourier transform Ill-posed problems Heat equation Time-dependent coefficient Introduction There are several important ill-posed problems for parabolic equations A classical example is the backward heat equation In other words, it may be possible to specify the temperature distribution at a particular time t < T from the temperature 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 ½0; pŠ  ½0; TŠ such that ut ðx; tÞ ¼ aðtÞuxx ðx; tÞ; uð0; tÞ ¼ uðp; tÞ ¼ 0; ux; Tị ẳ gxị; x; tị ẵ0; p  ð0; TŠ; t ½0; TŠ; x ½0; pŠ; ð1Þ ð2Þ ð3Þ 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 numerical calculations Hence, a regularization is in order In the special case of the problem (1)(3) with atị ẳ 1, the problem becomes ut x; tị ẳ uxx x; tị; x; tị ẵ0; p 0; T; u0; tị ¼ uðp; tÞ ¼ 0; uðx; TÞ ¼ gðxÞ; t ẵ0; T; x ẵ0; p: 4ị 5ị 6ị 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 ⇑ Corresponding author E-mail address: lmtriet2005@yahoo.com.vn (T Le Minh) 0096-3003/$ - see front matter Ó 2012 Elsevier Inc All rights reserved http://dx.doi.org/10.1016/j.amc.2012.11.069 T Nguyen Huy et al / Applied Mathematics and Computation 219 (2013) 6066–6073 6067 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 ut x; tị ẳ atịuxx x; tị; u0; tị ẳ up; tị ẳ 0; x; tị ẵ0; p 0; T; 7ị t ẵ0; T; ux; Tị ỵ bux; 0ị ẳ gxị; 8ị x ẵ0; pŠ: ð9Þ By applying the Fourier method, we can find the form of the solution of (7)–(9) n o Rt exp Àm2 aðsÞds n o g m sinðmxÞ; uðx; tị ẳ R T asịds mẳ1 b ỵ exp m X x; tị ẵ0; p ½0; TŠ: ð10Þ The regularized solution (13) based on modifying the solution (10) of the problem (7)–(9) (noting that when a ¼ 0, the solution (13) is the solution (10)) In this paper, we use the regularized solution (13) directly In Theorem 2.2, we can get the error estimate of Holder type for all t by using an appropriate parameter a P In fact, the error estimate for the case < t < T is as follows p2 tỵpa ku; tị v  ; tịk þ A1 Þq2 Tþqa : In this case, we can choose a ¼ and require a soft condition of the exact solution u A1 ẳ ku; 0ịk < 1: On the other hand, the error estimate for the case t ẳ is as follows pa ku; tị v  ; tịk ỵ A1 ịq2 Tỵqa : In order to get the the error estimate of Holder type, we choose a > and require a strong condition of the exact solution u A1 ¼ pX m¼1 2 È É exp 2m2 a um ð0Þ !12 < 1: It requires the exact solution u is smooth enough The remainder of the paper is divided into two sections In Section 2, we establish the regularized solution and estimate the error between an exact solution u of problem (1)–(3) and the regularized solution u with the Holder type Finally, a numerical experiment will be given in Section Regularization and main results We denote that k Á k is the norm in L2 ð0; pÞ Let hÁ; Ái be the inner product in L2 ð0; pÞ and g  be the measured data satisfying kg  ị gịk  Let atị : ẵ0; TŠ ! R be the differentiable function for every t and satisfy < p aðtÞ q; t T: ð11Þ Suppose that Problems (1)–(3) have an exact solution u then u can be formulated as follows n o Rt exp Àm2 aðsÞds X n o g m sinmxị; ux; tị ẳ R T asịds mẳ1 exp m x; tị ẵ0; p ẵ0; T: 12ị Let b > and a P 0, we shall approximate the solution of the backward heat problem (1)–(3) by the regularized solution as follows n R o t exp m2 asịds ỵ a n R o g m sinmxị; v x; tị ẳ T asịds ỵ a mẳ1 b ỵ exp m X x; tị ẵ0; p ẵ0; TŠ: We note that b depends on e such that lim beị ẳ and a is an arbitrarily nonnegative number e!0 Next, we consider some lemmas which is useful to the proof of theorems ð13Þ 6068 T Nguyen Huy et al / Applied Mathematics and Computation 219 (2013) 6066–6073 Lemma 2.1 Let x R; c > 0, a b, and b–0 then exa a cb : ỵ cexb 14ị Proof of Lemma 2.1 We have exa exa exa Àab ¼ : a a a c xb 1 ỵ ce þ cexb Þb ð1 þ cexb Þ b ð1 þ cexb Þb This completes the proof of Lemma 2.1 h Lemma 2.2 Let aðtÞ satisfy (11) and < b < Then for m P 1, one has n R o t exp m2 asịds ỵ a qtTị n R o b pTỵa T b ỵ exp m asịds ỵ a for all a P Proof of Lemma 2.2 From Lemma 2.1, we obtain n R o t  cðtÞ exp Àm2 aðsÞds þ a n R o ; T b b ỵ exp m asịds ỵ a RT where ctị ẳ Rt asịds RT asịds asịdsỵa From (11), we get Z T asịds P Z T aðsÞds t Z Z T pds ¼ pT; T qds ¼ qðT À tÞ: t Hence we get cðtÞ qðTÀtÞ Thus, since (15) gives pTỵa n R o t  qTtị exp m2 asịds ỵ a qtTị pTỵa n R o ẳ b pTỵa : T b b ỵ exp m asịds ỵ a This completes the proof of Lemma 2.2 h Lemma 2.3 Let aðtÞ satisfy (11) and < b < Then for m P 1, one has n R o t b exp Àm2 asịds ỵ a ptỵa n R o bqTỵa T b ỵ exp m asịds ỵ a for all a P Proof of Lemma 2.3 From (15), we have n R o t  cðtÞ b exp m2 asịds ỵ a n R o b ẳ b1ctị : T b b ỵ exp m asịds ỵ a From (11), we get n R o t b exp m2 asịds ỵ a ptỵa n R o bqTỵa : T b ỵ exp m asịds ỵ a This completes the proof of Lemma 2.3 h ð15Þ T Nguyen Huy et al / Applied Mathematics and Computation 219 (2013) 6066–6073 6069 Theorem 2.1 Let < b < and a P If v and w in Y are defined by (13) corresponding to the final values g and h in L2 ð0; pÞ, respectively then qðtÀTÞ kv ðÁ; tị w; tịk b pTỵa kg hk: Proof of Theorem 2.1 From ð16Þ v and w are defined by (13) corresponding to the final values g and h in L2 ð0; pÞ, we have n R o t exp m2 asịds ỵ a n R o g m sinmxị; v x; tị ẳ T asịds ỵ a mẳ1 b ỵ exp m x; tị ẵ0; p ẵ0; T 17ị n R o t exp m2 asịds ỵ a n R o hm sinmxị; wx; tị ẳ T asịds ỵ a mẳ1 b ỵ exp m x; tị ẵ0; p ẵ0; T; 18ị X and X where gm ẳ Z p p gxị sinmxịdx; hm ẳ Z p p hxị sinmxịdx: 19ị By using Lemma 2.2, we obtain n  R o 2 t   exp m2 asịds ỵ a X  2qðtÀTÞ pX p 2qðtÀTÞ 2   n   o kv ðÁ; tÞ À w; tịk ẳ h ị jg m hm j ẳ b pTỵa kg hk : b pTỵa ðg m  R m  T m¼1b ỵ exp m2  asịds ỵ a mẳ1 ð20Þ Therefore, we get qðtÀTÞ kv ðÁ; tÞ À w; tịk b pTỵa kg hk: 21ị This completes the proof of Theorem 2.1 h p Theorem 2.2 Let b ¼ q ; a P and g; g  L2 ð0; pÞ satisfy kg À g  k  If we suppose that u is the solution of problem (1)–(3) such that A1 ¼ pX mẳ1 ẩ ẫ exp 2m2 a jum 0ịj2 !12 < 1; ð22Þ then one has for every t ẵ0; T p2 tỵpa ku; tị v  ; tịk ỵ A1 ịq2 Tỵqa ; where v  ðx; tÞ ð23Þ is defined by (13) corresponding to the noisy data g  ðxÞ Proof of Theorem 2.2 Let u be defined by (13) with exact data g Using the triangle inequality, we get kuðÁ; tÞ À v  ðÁ; tÞk kuðÁ; tÞ À u ðÁ; tịk ỵ ku ; tị v  ; tịk: ð24Þ For the term ku ðÁ; tÞ À v  ðÁ; tÞk, using (16), we estimate it as follows qðtÀTÞ qðtÀTÞ ku ðÁ; tÞ À v  ðÁ; tÞk b pTỵa kg  ị gịk b pTỵa : ð25Þ From (12), we get the mth Fourier sine coefficient of u n o Rt exp Àm2 aðsÞds n o gm : um tị ẳ RT exp m2 aðsÞds Since (13), we get ð26Þ 6070 T Nguyen Huy et al / Applied Mathematics and Computation 219 (2013) 60666073 n R o t exp m2 asịds ỵ a n R o g m : um tị ẳ T b ỵ exp m2 asịds ỵ a  ð27Þ From (26) and (27) and using Lemma 2.3, we obtain n o n R o  0 R t   t exp m2 asịds ỵ a     exp Àm aðsÞds   um tị u tị ẳ @ A n o n   o g À RT RT m m    exp m2 asịds b ỵ exp m2 asịds ỵ a n R o t b exp m2 asịds ỵ a n R o n R o jg m j ¼ T T b þ exp Àm2 aðsÞds þ a exp Àm2 asịds ỵ a n R o t b exp m2 asịds ỵ a ẩ ẫ ẩ ẫ ptỵa n R o exp m2 a jum 0ịj bqTỵa exp m2 a jum 0ịj: ẳ T b ỵ exp m asịds ỵ a 28ị This follows that ku; tị u ; tịk ẳ  pX  mẳ1 2 2ptỵaị p X ẩ ẫ um tị um tị b qTỵa exp 2m2 a jum 0ịj2 : mẳ1 Hence, we obtain ptỵa ku; tị u ; tịk A1 bqTỵa : where A21 ẳ p P1 mẳ1 ẩ 29ị ẫ exp 2m2 a jum ð0Þj2 p From (24), (25), (29) and b ẳ q , we have ptỵa qtTị ku; tị v  ; tịk A1 bqTỵa þ b pTþa  A1 ptþa  p qTþ a  q ỵ  p qtTị pTỵa  q p2 tỵpa q2 Tỵqa  A1  ptỵa p2 tỵpa ỵ qTỵa ỵ A1 ịq2 Tỵqa : This completes the proof of Theorem 2.2 h Numerical experiment Consider the linear homogeneous parabolic equation with time-dependent coefficient x; tị ẵ0; p 0; 1; ut x; tị ẳ atịuxx x; tị; u0; tị ẳ up; tị ẳ 0; t ẵ0; 1; where atị ẳ 2t ỵ 30ị and ux; 1ị ẳ g ex xị ¼ sin x : e2 Hence, we obtain kg ex k ¼  !12 rffiffiffiffi Z p sin x2 p À2    e2  ds ¼ e and aðtÞ for all t ½0; 1Š The exact solution of the equation is ð31Þ 6071 T Nguyen Huy et al / Applied Mathematics and Computation 219 (2013) 6066–6073 n o Rt exp Àm2 aðsÞds X n o g m sinðmxÞ; uðx; tị ẳ R asịds mẳ1 exp m Rp where g m ẳ p2 32ị g ex xị sinmxịdx Let t ẳ 0, from (32), we have ux; 0ị ẳ X g sinmxị: 2m2 m e mẳ1 Consider the measured data e ! ỵ pffiffipffi À2 g ex ðxÞ; e g e ðxÞ ¼ ð33Þ then we have  !12 Z p   sin x2 e     g eÀgex  ¼ ppffiffiffi À2  e2  dx ¼ e: e Let a ¼ 0, from (13) and (33), we have the regularized solution for the case t ¼ v e ðx; 0Þ ¼ X m¼1 where g me ¼ p2 Let Rp g sinmxị; b ỵ e2m2 me g e xị sinmxịdx and a ¼ e be e1 ¼ 10À1 ; e2 ¼ 10À2 ; e3 ¼ 10À3 ; e4 ¼ 10À4 ; e5 ¼ 10À5 and b ¼ e3 , respectively Let Re tị ẳ   v e ;tịu;tị  i kuðÁ; tÞk 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 and t ¼ 0:1 We have the following table for the case t ẳ  v e ;0ịu e i À1 e1 ¼ 10 e2 ¼ 10À2 e3 ¼ 103 e4 ẳ 104 e5 ẳ 105 ex ;0ị   Re ð0Þ 6.411579eÀ001 5.1157575999eÀ001 5.914401eÀ001 4.7190616771eÀ001 4.215352eÀ001 3.3634022181eÀ001 2.549425eÀ001 2.0341697917eÀ001 1.372796eÀ001 1.0953450889eÀ001 We have the following table for the case t ¼ 0:1 e e1 ¼ 10À1 e2 ¼ 10À2 e3 ¼ 10À3 e4 ¼ 10À4 e5 ¼ 10À5  v e ðÁ;0:1ÞÀu i ex ðÁ;0:1Þ   Re ð0:1Þ 5.743711eÀ001 5.1155245814eÀ001 5.298322eÀ001 4.7188475240eÀ001 3.776257eÀ001 3.3632499109eÀ001 2.283862eÀ001 2.0340773067eÀ001 1.229797eÀ001 1.0952947987e001 where kuex ; 0ịk ẳ 21=2ị p1=2ịị=2 1:2533 and kuex ; 0ịk ẳ p1=2ị 1=2 expð11=50ÞÞÞð1=2Þ ’ 1:1228 We have the following graph of the exact solution uex ðÁ; tÞ and of the regularized solution v ei ; tị; i ẳ 1; 2: 6072 T Nguyen Huy et al / Applied Mathematics and Computation 219 (2013) 6066–6073 We have the following graph of the regularized solution v e ; tị; i ẳ 3; 4; 5: i Now, the figure can represent visually the exact solution and the regularized solution at initally time t = Now, the figure can represent visually the exact solution and the regularized solution at the time t = 0.1 T Nguyen Huy et al / Applied Mathematics and Computation 219 (2013) 6066–6073 6073 Acknowledgement 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 [3] R.E Ewing, The approximation of certain parabolic equations backward in time by Sobolev equations, SIAM J Math Anal (2) (1975) 283–294 [4] D.N Hao, A mollification method for ill-posed problems, Numer Math 68 (1994) 469–506 [5] T Schroter, U Tautenhahn, On optimal regularization methods for the backward heat equation, Z Anal Anw 15 (1996) 475–493 [6] D Lesnic, L Elliott, D.B Ingham, An iterative boundary element method for solving the backward heat conduction problem using an elliptic approximation, Inv Prob Eng (1998) 255–279 [7] B Yildiz, M Ozdemir, Stability of the solution of backward heat equation on a weak compactum, Appl Math Comput 111 (2000) 1–6 [8] N.S Mera, L Elliott, D.B Ingham, D Lesnic, An iterative boundary element method for solving the one dimensional backward heat conduction problem, Int J Heat Mass Trans 44 (2001) 1937–1946 [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 [12] B Yildiz, H Yetis, A Sever, A stability estimate on the regularized solution of the backward heat problem, Appl Math Comput 135 (2003) 561–567 [13] Y Huang, Q Zhneg, Regularization for ill-posed Cauchy problems associated with generators of analytic semigroups, J Differ Equ 203 (1) (2004) 38–54  ... 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