Applied Mathematics and Computation 216 (2010) 3423–3432 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc The truncation method for a two-dimensional nonhomogeneous backward heat problem Phan Thanh Nam a, Dang Duc Trong b, Nguyen Huy Tuan c,* a Department of Mathematical Sciences, University of Copenhagen, Denmark Faculty of Mathematics and Computer Sciences, University of Science, Vietnam National University, HoChiMinh City, Viet Nam c Faculty of Mathematics, SaiGon University, HoChiMinh City, Viet Nam b a r t i c l e i n f o a b s t r a c t We consider the backward heat problem Keywords: Backward heat problem Ill-posed problem Nonhomogeneous heat Truncation method Error estimate ut À uxx À uyy ¼ f x; y; tị; ux; y; Tị ẳ gx; yị; ðx; y; tÞ X  ð0; TÞ; ðx; yÞ X; with the homogeneous Dirichlet condition on the rectangle X = (0, p)  (0, p), where the data f and g are given approximately The problem is severely ill-posed Using the truncation method for Fourier series we propose a simple regularized solution which not only works on a very weak condition on the exact data but also attains, due to the smoothness of the exact solution, explicit error estimates which include the approximation pffiffiffi ðlnðÀ1 ÞÞ3=2 in H2(X) Some numerical examples are given to illuminate the effect of our method Ó 2010 Elsevier Inc All rights reserved Introduction Let X = (0, 1)  (0, 1) be a heat conduction rectangle Given the heat source f(x, y, t) on (x, y, t) X  [0, T] and the final temperature u(x, y, T) at some time T > 0, we consider the problem of recovering the temperature distribution u(x, y, t) from the backward heat problem ut À uxx À uyy ¼ f ðx; y; tÞ; ðx; y; tÞ X  ð0; TÞ; uð0; y; tị ẳ up; y; tị ẳ ux; 0; tị ¼ uðx; p; tÞ ¼ 0; uðx; y; TÞ ¼ gðx; yÞ; ð1Þ t ð0; TÞ; ðx; yÞ X: ð2Þ ð3Þ Since f and g come from measurement, they are in general non-smooth and only approximate values This is a typical example of the inverse and ill-posed problem since although this problem has at most one solution (see Theorem in Section 2), the solution does not always exist, and in the case of existence, it does not depend continuously on the given data The instability makes the numerically calculus difficult and hence a regularization is in order The homogeneous backward heat problems, i.e the case f = 0, was extensively considered by many authors using many approach, e.g the original quasi-reversibility method of Lattès and Lions [10], the quasi-boundary value problem method [15], the quasi-solution method of Tikhonov and Arsenin [16], the logarithmic convexity method [1] and the C-regularized semi-groups technique [7] Physically, this problem arises from the requirement of recovering the heat temperature at some * Corresponding author E-mail address: tuanhuy_bs@yahoo.com (N.H Tuan) 0096-3003/$ - see front matter Ó 2010 Elsevier Inc All rights reserved doi:10.1016/j.amc.2010.03.038 3424 P.T Nam et al / Applied Mathematics and Computation 216 (2010) 3423–3432 earlier time using the knowledge about the final temperature The problem is also involved to the situation of a particle moving in a environment with constant diffusion coefficient (see [6]) when one asks to determine the particle position history from its current place The interest of backward heat equations also comes from financial mathematics, where the celebrated Black–Scholes model [2] for call option can be transformed into a backward parabolic equation whose form is related closely to backward heat equations Although there are many papers on the homogeneous backward heat equation, the result on the inhomogeneous case is very scarce while the inhomogeneous case is, of course, more general and nearer to practical application than the homogeneous one Shortly, it allows the appearance of some heat source which is inevitable in nature Let us mention here some approaches and their technical difficulties of many earlier works In the method of quasireversibility, the main ideas is of replacing the unbounded operator A (in our case is ÀD) by a perturbed one A In the original method in 1967, Lattès and Lions [10] proposed A(A) = A À eA*A, i.e adding a ‘‘corrector” into the original operator, to obtain a well-posed problem The essential difficulty of the quasi-reversibility method is due to the appearance of the second-order operator A*A which produces serious difficulties on the numerical implementation In addition, the stability magT nitude of the approximating problem, i.e the error introduced by a small change in the final value, is of order e which is very large when becomes small In 1983, Showalter [15] presented the quasi-boundary value method for the homogeneous problem which gave a stability estimate better than the one of the quasi-reversibility method discussed above The main ideas of this method is of adding an appropriate ‘‘corrector” into the final data (instead of the main equation) Using this method, Clark and Oppenheimer [3], and very recently Denche and Bessila [4], regularized the backward heat problem by replacing the nal condition by uTị ỵ u0ị ẳ g and uTị u0 0ị ẳ g; respectively This method, in general, gives the stability estimate of order À1 Although there are many papers on the homogeneous case of the backward problem, we only find a few result on the inhomogeneous case, and especially the two-dimensional case is very scarce In 2006, Trong and Tuan [17] approximated a one-dimensional inhomogeneous linear problem by the quasi-reversibility method As we mention before, the stability T magnitude of the method is of order e In their work the error between the approximate problem and the exact solution is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u @ f ðx; tÞ2 u8 ; T tịt ku:; 0ịk2 ỵ t2 @x4 t L ð0;T;L ð0;pÞÞ which is very large when t becomes small In 2007, Trong et al [19] used an improved version of quasi-boundary value method to regularize the one-dimensional version of (1)–(3) for a nonlinear heat source f = f(x, t, u) Their error estimate is t/T for t > and (ln(1/))1/4 for t = One of the essential requirements of the previous works on inhomogeneous problem, e.g [17,19], is X e2Tk g 2k < 1; 4ị kẳ1 where gk is the coefficient of the Fourier series of the final datum u(.,T) = g, i.e gk ¼ Z p p gðxÞ sinðkxÞdx: While such a condition is reasonable in homogeneous problems, it is not necessarily true in the inhomogeneous case For example, consider the problem ut À uxx ¼ f ðx; tÞ et x; ðx; tÞ ð0; pị 0; Tị; u0; tị ẳ up; tị ẳ 0; t ð0; TÞ: Corresponding to the final value u(x,1) = g(x) ex, the equation has a (unique) solution u(x,t) = etx However, by direct comkỵ1 putation we nd that g k ẳ 2e 1ịk and hence X e2Tk g 2k ¼ 4e2 k¼1 X e2Tk k¼1 k > 1: In the present paper, we not need condition (4) In fact, we shall give a simple and convenient way to construct the regularized method which works with very weak assumption on the exact solution Let us give a simple analysis for the ill-posedness of the problem (1)–(3) This problem may be rewritten formally as ux; y; tị ẳ X m;nẳ1 eTtịm ỵn2 ị Z g mn t T esTịm ỵn2 ị fmn sịds sinmxị sinnyị; 5ị 3425 P.T Nam et al / Applied Mathematics and Computation 216 (2010) 3423–3432 where gmn and fnm(t) are the coefficient of the Fourier-sin expansion of g and f(., , t), i.e g mn :ẳ Z gx; yị sinmxị sinnyịdxdy; Z fmn tị :ẳ f x; y; tị sinmxị sinnyịdxdy: p2 p X X Ttịm2 ỵn2 ị 2 If t < T then e increases very fast when m2 + n2 becomes large Thus the term eÀðtÀTÞðm þn Þ is the source of instability It is a natural think to recover the stability of problem (5) is to filter all high frequencies In the present paper, we simply that by using the truncated regularization method, namely taking the sum (5) only for m2 + n2 M with an appropriate regularization parameter M The truncated regularization method is a very simple and effective method for solving some illposed problems and it has been successfully applied to some inverse heat conduction problems [5,8,13] However, in many earlier works, we find that only logarithmic type estimates in L2-norm are available; and estimates of Hölder type are very rate (see Remarks and for more detail comparisons) In our method, corresponding to different levels of the smoothness of the exact solution, the convergence rates will be improved gradually In particular, if we impose a condition similar to (4) pffiffiffi then the error estimate in H2(X) is ðlnðÞÞ3=2 , which is better than any Hölder estimate of order q with q (0, 1/2) We mention that our regularized solution in all case is unique, and all error estimates are valid for all t [0, T] The remainder of the paper is organized as follows In Section we shall construct the regularized and show that it works even with very weak condition on the exact solution In Section 3, many error estimates are derived, in both of the usual cases such as the exact solution u in H10 ðXÞ or H2(X), and the special cases when the exact solution is very smooth Some numerical experiments are given in Section to illuminate the effect of our method Regularized solution Let us first make clear what a weak solution of the problem (1)–(3) is As follows we shall write u(t) = u(., , t) for short We call a function u C([0, T]; L2(X)) \ C1((0, T); L2(X)) to be a weak solution for the problem (1)–(3) if d huðtÞ; WiL2 ðXÞ À huðtÞ; DWiL2 ðXÞ ¼ huðtÞ; DWiL2 ðXÞ ; dt ð6Þ for all function Wðx; yÞ H2 ðXÞ \ H10 ðXÞ In fact, it is enough to choose W in the orthogonal basis {sin(mx)sin(ny)}m,nP1 and the formula (6) reduces to umn tị ẳ eTtịm ỵn2 ị g mn Z T estịm ỵn2 ị fmn sịds; 8m; n P 1; 7ị t which may also be written formally as (5) Note that if the exact solution u is smooth then the exact data (f, g) is smooth also However, the real data, which come from practical measure, is often discrete and non-smooth We shall therefore always assume that f L1(0, T; L1(X)) and g L1(X), and the error of the data is given on L1 only Note that (7) still makes sense with such data, and this formula gives immediately the uniqueness Theorem (Uniqueness) For each f L1(0, T; L1 (X)) and g L1 (X), the problem (1)–(3) has at most one (weak) solution u C([0, T]; L2(X)) \ C1((0, T); L2(X )) In spite of the uniqueness, the problem is still ill-posed and a regularization is necessary For each > 0, introduce the truncation mapping P : L1 ðXÞ ! C Xị \ H10 Xị X P wx; yị ẳ wmn sinmxị sinnyị; with M ẳ m;nP1;m2 ỵn2 6M ln1 Þ : 2T ð8Þ In fact, P is a finite-dimensional orthogonal projection on L2(X), but it works on L1(X) as well We shall approximate the original problem by the following well-posed problem Theorem (Well-posed problem) For each f L1(0, T; L1(X)) and g L1(X), let w L1(0, T; L2(X)) dened by ỵn2 ị wmn tị ẳ eTtịm P gịmn Z t T ỵn2 Þ eðsÀtÞðm ðP f Þmn ðsÞds; 8m; n P 1: ð9Þ Then w = Pw and it depends continuously on (f, g), i.e if wi is the solution with respect to (fi, gi), i = 1, 2, then kw1 ðtÞ À w2 ðtÞkL2 ðXÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lnðeÀ1 Þ tÀT pffiffiffiffiffiffi e 2T kg À g kL1 Xị ỵ kf1 f2 kL1 0;T;L1 Xịị : p 2T Proof Note that w(t) is well-defined because wmn(t) = if m2 + n2 > M This fact also implies that w = P w Now for two solutions w1, w2 we have 3426 P.T Nam et al / Applied Mathematics and Computation 216 (2010) 3423–3432 kw1 ðtÞ w2 tịk2L2 Xị ẳ ẳ 6 p2 X p2 4 p2 p jw1;mn ðtÞ w2;mn tịj2 m;nP1 X m;nP1;m2 ỵn2 6M X m;nP1;m2 ỵn2 6M Z Ttịm2 ỵn2 ị e ðg À g Þmn À T eðsÀtÞðm t Z ðTÀtÞM e e kg À g kL1 Xị ỵ T TtịM e e t ỵn2 ị 2 f1 f2 ịmn ðsÞds 2 kf1 ðsÞ À f2 ðsÞkL1 ðXÞ ds 2 M e e2ðTÀtÞMe kg À g kL1 Xị ỵ kf1 f2 kL1 0;T;L1 Xịị : Here we have used jv mn j jv jL1 ðXÞ and the fact that #fðm; nÞ Z2 jm; n P 1; m2 ỵ n2 M e g Me : Reviewing the value of Me, we have the desired estimate h Remark A significant convenience of our method is that it is very easy to compute and represent explicitly the solution w defined by (9) Moreover, this solution is very smooth because wtị ẳ P wtị C ðXÞ \ H10 ðXÞ for all t [0, T] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tÀT Remark The stability magnitude of our well-posed problem is of order lnðeÀ1 Þe 2T It is much better, especially when t = 0, than the stability magnitudes given by quasi-reversibility method and quasi-boundary value method, for example, tÀTT in [3,19] and (ln(À1))À1 in [4,18] Our regularized solution is the solution produced directly by the well-posed problem in the previous section from the given data which works even on a very weak assumption on the exact solution Theorem (Regularized solution) Assume that the problem (1)–(3) has at most one (weak) solution u C([0, T]; L2(X)) \ C1((0, T); L2(X)) corresponding to f L1(0, T; L1(X)) and g L1(X) Let f and g be measured data satisfying kf À f kL1 ð0;T;L1 ðXÞÞ ; kg À gkL1 ðXÞ : Define the regularized solution u L1(0,T; L2 (X)) from f and g as in (9) Then for each t ẵ0; T; u tị C Xị \ H10 ðXÞ and lime?0u(t) = u(t) in L2(X) Proof We shall use the notations P and M defined in (8) Note that u tị ẳ P u tị C ðXÞ \ H10 ðXÞ as in Remark Moreover using the stability in Theorem we find that kue ðtÞ À uðtÞkL2 ðXÞ kP ue ðtÞ Pe utịkL2 Xị ỵ kPe utị utịkL2 Xị 11=2 p X lne1 ị Tỵt p @ 2A p e 2T ỵ jumn tịj m;nP1;m2 þn2 >M p 2T e ð10Þ and it must converge to as ? To obtain the convergence of the second term in the right-hand side of (10), we note that p2 X jumn tịj2 ẳ kutịk2L2 ðXÞ < m;nP1 and M ? as ? h In the above theorem, we did not give an error estimate because the condition of the exact solution u is so weak (we even did not require uðtÞ H10 ðXÞ) However in practical application we may expect that the exact solution is smoother In these cases many explicit errors estimates are available in the next section An essential point here is that the regularized solution is the same in any case This is a substantial pleasure for practical application because even if someones not know how good the exact solution is they are always ensured that the regularized solution works as well as possible without any further adjustment Error estimates From the usual viewpoint from variational method, it is natural to assume that uðtÞ H10 ðXÞ for all t [0, T] Moreover, if f is smooth and u is a classical solution for the heat Eq (1) then uðtÞ H2 ðXÞ \ H10 ðXÞ for all t [0, T] For these two cases we have the following explicit error estimates Theorem (Error estimate for usual cases) Let u, u as in Theorem and let t [0, T] 3427 P.T Nam et al / Applied Mathematics and Computation 216 (2010) 3423–3432 (i) Assume that uðtÞ H10 ðXÞ Then lim u tị ẳ utị in H10 Xị and kue ðtÞ À uðtÞkL2 ðXÞ pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !0 2T lne1 ị Tỵt p e 2T ỵ p krutịkL2 Xị : lnðeÀ1 Þ p 2T (ii) Assume that uðtÞ H2 ðXÞ \ H10 ðXÞ Then lim?0u(t) = u(t) in H2(X) and p lne1 ị Tỵt 2T p e 2T ỵ kue tị utịkL2 Xị kutịkH2 ðXÞ ; lnðeÀ1 Þ p 2T pffiffiffiffiffiffi lnðeÀ1 Þ Tỵt 2T kutịk2H2 Xị : e 2T ỵ p kue ðtÞ À uðtÞkH1 ðXÞ pT lnðeÀ1 Þ Here we use the norm kwk2H1 ¼ krwk2L2 ¼ kwx k2L2 þ kwy k2L2 ; kwk2H2 ¼ kwk2L2 þ kwk2H1 þ kwxx k2L2 þ kwxy k2L2 þ kwyx k2L2 þ kwyy k2L2 : Proof (i) By using the integral by part and the Parseval equality, it is straightforward to check that if uðtÞ H10 ðXÞ then p2 X m2 ỵ n2 ịjumn tịj2 ẳ krutịk2L2 Xị : ð11Þ m;nP1 Using (11) we have X jumn ðtÞj2 m;nP1;m2 ỵn2 >Me X m2 ỵ n2 ịjumn tịj2 ẳ krutịk2L2 Xị : M e m;nP1 p Me Substituting the latter inequality into the estimate (10) in the proof of Theorem 3, we obtain the error estimate in L2 To prove the convergence in H10 we use the identity (11) and the stability of Theorem again krue tị rutịk2L2 Xị ẳ ẳ p2 X m;nP1 X p2 m2 ỵ n2 ịjue;mn tị umn tịj2 m;nP1;m2 ỵn2 6M Me jPe ue ịmn tị Pe uịmn tịj ỵ m;nP1;m2 ỵn2 6M e Me kPe ue tị Pe utịk2L2 Xị ỵ 4lne1 ịị2 p 2T2 Tỵt eT ỵ p2 X p2 X p2 e X p2 m2 ỵ n2 ịjue;mn tị umn tịj2 ỵ m;nP1;m2 ỵn2 >M e X p2 m2 ỵ n2 ịjumn tịj2 m2 ỵ n2 ịjumn tịj2 m;nP1;m2 ỵn2 >M e m2 ỵ n2 ịjumn tịj2 m;nP1;m2 ỵn2 >M e X m2 ỵ n2 ịjumn tịj2 : m;nP1;m2 ỵn2 >M ð12Þ e The second term in the right-hand side in (12) converges to as ? because the convergence in (11) Thus the convergence in H10 has been proved (ii) We now assume that uðtÞ H2 ðXÞ \ H10 ðXÞ We have an identity similar to (11) p2 X m2 ỵ n2 ị2 jumn tịj2 ẳ kuxx tịk2L2 Xị ỵ kuxy tịk2L2 Xị ỵ kuyx tịk2L2 Xị ỵ kuyy tịk2L2 Xị : m;nP1 The error estimate in L2(X) follows (10) and the following inequality X m;nP1;m2 þn2 >Me jumn ðtÞj2 M 2e X ðm2 þ n2 Þ2 jumn ðtÞj2 m;nP1 p2 M2e kuðtÞk2H2 ðXÞ : Similarly, from (12) and the estimate X m;nP1;m2 ỵn2 >Me we nd that m2 ỵ n2 ịjumn tịj2 X m2 ỵ n2 ị2 jumn ðtÞj2 kuðtÞk2H2 ðXÞ ; M e m;nP1 p Me ð13Þ 3428 P.T Nam et al / Applied Mathematics and Computation 216 (2010) 3423–3432 krue ðtÞ À ruðtÞk2L2 ðXÞ p2 lne1 ịị2 4T Tỵt eT ỵ kuðtÞk2H2 ðXÞ : Me pffiffiffi pffiffiffi2 Using the inequality a þ b a þ b we obtain the error estimate in H10 Finally we prove the convergence in H2(X) Similarly to (12) we have kðue À uÞxx ðtÞk2L2 þ kðue À uÞxy ðtÞk2L2 þ kðue À uÞyx ðtÞk2L2 þ kðue À uÞyy ðtÞk2L2 p2 X 2 ẳ m ỵ n ị jue;mn tị umn tịj2 m;nP1 X p2 M 2e jue;mn ðtÞ umn tịj2 ỵ m;nP1;m2 ỵn2 6M M e kPe ue ðtÞ À as 2ðlnðeÀ1 ÞÞ3 p2 T e P e utịk2L2 Xị Tỵt T e þ 4 m;nP1;m2 þn2 >M X m;nP1;m2 þn2 >M X p2 X p2 ỵ p2 m2 ỵ n2 ị2 jumn tịj2 e 2 m2 ỵ n ị jumn tịj2 e m2 ỵ n2 ị2 jumn tịj2 ! 14ị m;nP1;m2 ỵn2 >Me ? due to the convergence in (13) h Remark In Theorem we have pointwise estimates due to the pointwise condition on the exact solution u As a consequence, we shall immediately obtain a uniform convergence whenever the corresponding uniform condition is imposed For example, if the exact solution u is in Cẵ0; T; H10 Xịị or Cẵ0; T; H2 Xị \ H10 Xịị then we have the estimates kue ukCẵ0;T;L2 Xịị and kue ukCẵ0;T;H1 Xịị , respectively, in the same form of estimates in Theorem Remark The error estimates in Theorem work well no matter t > or t = In many earlier works, we find that the error ku ð0Þ À uð0ÞkL2 ðXÞ is often not given (e.g [17]) and an explicit error estimate in H10 ðXÞ is not available (e.g [3,7,4,17,19,18]) In Theorem (ii), an error estimate in H2(X) is not given because we not have enough information on the exact solution (we just know uðtÞ H2 ðXÞ \ H10 ðXÞ) However, when u is smoother then an explicit estimate in H2(X) may be derived In the last theorem, we shall give the error estimates in some special cases when the exact solution is very good We see from the proof of Theorem that the facts uðtÞ H10 ðXÞ and uðtÞ H2 ðXÞ \ H10 Xị are equivalent to X m2 ỵ n2 ịk jumn ðtÞj2 < m;nP1 with k = 1,2, respectively We shall see that from the latter condition with k > we may improve the estimate, and in particular give an error estimate in H2(X) We next consider a stronger condition similar to (in fact, weaker than) sup X ỵn2 ị e2Tm jumn tịj2 < 1; 15ị t2ẵ0;T m;nP1 which is a two-dimensional version of the condition (4) in [14] Such a condition seems essential to solve the nonlinear problem Although it is quite strict for the linear case, as we discussed in the first section, if the above condition (15) holds then pffiffiffi we have a very good convergence rate which is of order ðlnðÀ1 ÞÞ3=2 Theorem Error estimate for special casesLet u, u as in Theorem and let t [0, T] (i) Assume that Ek tị ẳ X n2 ỵ m2 ịk u2mn tị < n;mẳ1 for some constant k > Then p p 2k lne1 ị Tỵt p Ek tị 2T 2T p e ỵ kue tị À uðtÞkL2 ðXÞ ; À1 lnðe Þ p 2T p k1 2 lne1 ị Tỵt p Ek ðtÞ 2T kue ðtÞ À uðtÞkH1 ðXÞ e 2T ỵ ; lne ị pT p k2 2lne1 ịị3=2 Tỵt 3p Ek tị 2T pffiffiffi kue ðtÞ À uðtÞkH2 ðXÞ e 2T þ : À1 Þ lnð e pT T Here we assume eÀ2T for the estimate in H2(X) (ii) Assume that ð16Þ 3429 P.T Nam et al / Applied Mathematics and Computation 216 (2010) 3423–3432 X F r tị ẳ e2rm ỵn2 ị jumn tịj2 < m;nP1 for some constant r > Then pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi lne1 ị Tỵt p F r tị r p e 2T ỵ e2T ; p 2T p lne1 ị Tỵt p F r tị lne1 ị r p e 2T ỵ e2T ; pT 2T p 6lne1 ịị3=2 Tỵt 3p F r tị lne1 ị r p e 2T ỵ e2T : 4T pT T kue ðtÞ À uðtÞkL2 ðXÞ kue ðtÞ À uðtÞkH1 ðXÞ kue ðtÞ À uðtÞkH2 ðXÞ Here we assume eÀ2T for the estimate in H10 ðXÞ, and eÀ4T for the estimate in H2(X) Proof (i) We use the same way of the proof of Theorem We shall prove the error estimates in H2(X) (the other ones are similar and easier) From X m2 ỵ n2 ị2 jumn tịj2 m;nP1;m2 ỵn2 >Me X k2 Me m2 ỵ n2 ịk jumn tịj2 m;nP1 Ek ðtÞ M kÀ2 e and (14) we nd that 2lne1 ịị3 kue uịxx tịk2L2 ỵ kue uịxy tịk2L2 ỵ kue uịyx tịk2L2 ỵ kue À uÞyy ðtÞk2L2 6 2T3 p !2 pffiffiffiffiffiffiffiffiffiffiffi 3=2 Tỵt p Ek tị k2 Me e 2T ỵ : Me p Tỵt eT ỵ p2 Ek 4M k2 e Using kwkH2 kwkL2 ỵ kwkH1 ỵ q kwxx k2L2 ỵ kwxy k2L2 ỵ kwyx k2L2 þ kwyy k2L2 ð17Þ and M P we conclude the desired estimate in H2(X) (ii) From (10) and X m;nP1;m2 ỵn2 >M jumn tịj2 e2rMe X ỵn2 Þ e2rðm r jumn ðtÞj2 F r ðtÞeT m;nP1 e we get the error estimate in L (X) Note that the function n ´ en/n is increasing when n P Thus ỵn2 M m2 ỵ n2 ị Me e2rm eị when m2 ỵ n2 > M e P 1: It implies that X m2 ỵ n2 ịjumn tịj2 M e X ỵn2 M e2rðm eÞ r jumn ðtÞj2 Me F r ðtÞeT : m;nP1 m;nP1;m2 ỵn2 >Me H10 The error estimate in XÞ follows the above estimate and (12) Similarly, because the function n ´ en/n2 is increasing when n P 2, we nd that m2 ỵ n2 ị2 M 2e e2rm ỵn2 M eị if m2 ỵ n2 > Me P 2: It follows that X m;nP1;m2 ỵn2 >M m2 ỵ n2 ịjumn tịj2 M 2e e Thus (14) reduces to X m;nP1 e2rm ỵn2 M eÞ r jumn ðtÞj2 M2e F r ðtÞeT : 3430 P.T Nam et al / Applied Mathematics and Computation 216 (2010) 34233432 kue uịxx tịk2L2 ỵ kue uịxy tịk2L2 ỵ kue uịyx tịk2L2 ỵ kue uịyy tịk2L2 6 p Tỵt 2T M 3=2 e e ỵ !2 p p F r tị r M e e2T P p2 T Tỵt r e T ỵ M2e F r tịeT : Using (17) again and M P we conclude the error estimate in H2(X) Remark If (15) holds, i.e 2ðlnðeÀ1 ÞÞ3 2Tm2 ỵn2 ị jumn tịj2 m;nP1 e h < 1, then applying Theorem in the case r = T we get pffiffiffi C e; sup kue ðtÞ À utịkL2 Xị t2ẵ0;T p sup kue tị utịkH1 Xị C lne1 ị e; t2ẵ0;T sup kue tị utịkH2 Xị Clne1 ịị3=2 p e: t2ẵ0;T Notice that in [11], under a similar condition, Liu gave the error estimate (see Theorem 3.3, p 466) kg a À g kL2 ðXÞ C 1ÀtT0 Let t0 = 0, we get kg a À g kL2 ðXÞ C pffiffiffiffiffi Thus at t = 0, our method gave the same order error in L2-norm as the method of Liu [11] However, the strong point of our paper is that the error estimates in H10 ðXÞ or H2(X) established, and also of Hölder type (in fact, they are better than any estimate of order q with q (0, 1/2)) They are not given in [11] Remark The truncated regularization method is a very simple and effective method for solving some ill-posed problems and it has been successfully applied to some inverse heat conduction problems [5,8,13] Recently, in [14] many applications for a model of the Helmholtz equation are introduced and a Fourier method was applied for solving a Cauchy problem for the Helmholtz equation In [9], Fu and his group used the truncated method to solve the backward heat in the unbounded region and established the logarithmic order of the form kuð:; tÞ À ud;nmax k E 1ÀTt tTịs E 2T @ ln 1ỵ d ln Ed T ln Ed ỵ ln ln Ed !2s ÁÀs A: ð18Þ 2T And in [20], the authors gave the following estimates ÀaðTÀtÞ a ! 2T 2 : kwb;ab ð:; :; tÞ À uð:; :; tÞk bt=T ln expðk ðT À tị2 ị ỵ Q b; t; uị ln b b ð19Þ And in [21], Trong and Tuan only established the logarithmic form as follows kuð:; :; tÞ À u ð:; :; tịk C ỵ ln T 20ị Note that the errors (18)–(20) are the same order as Theorem (i) However,the logarithmic type estimate is, in general, much worse than any Hölder type estimate, i.e q for some q > In Theorem (ii) we also establish this type of estimates, which are not given in [9,20,21] It worth mentioning that our regularized solution is unique, in all cases This proves that our method is effective Remark Sometime, it is also important to consider the 2-D backward heat for a general two-dimensional domain, e.g [12] In this case to apply the truncation method, we need to consider the spectral problem of operator ÀD in this domain (with homogeneous Dirichlet boundary condition) However, this question is not always solvable explicitly and this is a disadvantage point of our method Numerical experiments In this section we give some numerical experiments for our method For simplicity, we shall recover the initial temperature at t = from the final data at T = 3431 P.T Nam et al / Applied Mathematics and Computation 216 (2010) 3423–3432 Example Consider the problem ut À Du ¼ f ðx; y; tÞ 3et sinðxÞ sinðyÞ; with the final condition ux; y; 1ị ẳ gx; yị e sinxị sinðyÞ: Problem (1)–(3) with exact data (f, g) has the exact solution ux; y; tị ẳ et sinxị sinyị: For any n = 1, 2, , let us take the measured data fn ¼ f ; g n x; yị ẳ gx; yị ỵ n sinnxị sinnyị: Then Problem (1)–(3) with measured data (f, g) has corresponding solution e n x; y; tị ẳ ux; y; tị ỵ en 1tị sinnxị sinnyị: u n We see that ! 0; n en ~ n ð:; :; 0ị u:; :; 0ịkL2 Xị ẳ ku ! þ1 n kg n ðx; yÞ À gðx; yÞkL1 ðXÞ ¼ It means that if n is large then a small error of data might cause a large error of solutions Therefore, the problem is really unstable and hence a regularization is necessary Using the regularization of Theorem correspoding e = 4/n, we see that when n > 4e4 then the regularized solution at t = is ue x; yị ẳ sinxị sinyị; which coincides the exact solution u(., , 0) In this example, our method works very well because the exact solution’s form is of a truncated Fourier series Example Consider the problem ut À Du ¼ f ðx; y; tÞ sinðxÞðty3 À pty2 À 6ty ỵ 6y ỵ 2t p 2pị with the nal condition ux; y; 1ị ẳ gx; yị 0: The exact solution of the latter equation is uðx; y; tị ẳ tị sinxịy2 p yị: For any n = 1, 2, , take the measured data fn ẳ f ; g n x; yị ¼ sinðnxÞ sinðnyÞ: 4n Then the disturbed solution is ~ n x; y; tị ẳ ux; y; tị ỵ u n2 ð1ÀtÞ e sinðnxÞ sinðnyÞ: 4n We see that ! 0; n en ~ n ð:; :; 0ị u:; :; 0ịkL2 Xị ẳ ! ỵ1: ku 4n kg n ðx; yÞ À gðx; yÞkL1 ðXÞ ¼ Table Errors between disturbed solution, regularized solution and exact solution ue ~ À u0 kL2 ku kue À u0 kL2 kue À u0 kH1 10 10À5 4sin(x)sin(y) sinðxÞ sinðyÞ À 32 sinðxÞ sinð2yÞ exp(999992) exp(1010) 2.388527 0.391677 5.506293 1.600311 10À9 4 sinðxÞ sinðyÞ À 32 sinxị sin2yị ỵ 27 sinxị sin3yị exp(1018) 0.315050 1.421075 1015 4 sinxị sinyị 32 sinxị sin2yị ỵ 27 sinðxÞ sinð3yÞ À 16 sinðxÞ sinð4yÞ exp(1030) 0.111858 0.738103 e ¼ 1n À3 3432 P.T Nam et al / Applied Mathematics and Computation 216 (2010) 3423–3432 Thus the problem in this case is also unstable We now compute the regularized solutions by using the regularization method introduced in the previous sections with e = 1/n The effect of our regularization is represented via Table 1, where we denote e :ẳ u e n :; :; 0ị the disturbed value (with = 1/n), and u0 :¼ u(., , 0) the exact by u :¼ u(., , 0) the regularized value at t = 0, u value We can see that while the errors between the disturbed solution and the exact solution is extremely large, the error between the regularized solution and the exact solution is acceptable, even in H10 -norm Acknowledgments Most part of the work was done when the first author was a student of University of Science, Vietnam National University at HoChiMinh City The second and the third authors are supported by the Council for Natural Sciences of Vietnam We thank the referees for constructive comments leading to the improved version of the paper References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] S Agmon, L Nirenberg, Properties of solutions of ordinary differential equations in Banach spaces, Commun Pure Appl Math 16 (1963) 121–139 F Black, M Scholes, The pricing of options and corporate liabilities, J Polit Econ 81 (3) (1973) G.W Clark, S.F Oppenheimer, Quasireversibility methods for non-well posed problems, Electron J Diff Eqns (1994) 1–9 M Denche, K Bessila, A modified quasi-boundary value method for ill-posed problems, J Math Anal Appl 301 (2005) 419–426 L Elden, F Berntsson, T Reginska, Wavelet and Fourier methods for solving the sideways heat equation, SIAM J Sci Comput 21 (6) (2000) 2187–2205 L.C Evans, Partial Differential Equations, American Mathematical Society, 1998 R.E Ewing, The approximation of certain parabolic equations backward in time by Sobolev equations, SIAM J Math Anal (1975) 283–294 C.L Fu, Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation, J Comput Appl Math 167 (2004) 449–463 C.L Fu, X.T Xiong, Z Qian, Fourier regularization for a backward heat equation, J Math Anal Appl 331 (1) (2007) 472–480 R Lattès, J.L Lions, Méthode de Quasi-réversibilité et Applications, Dunod, Paris, 1967 J.J Liu, Numerical solution of forward and backward problem for 2-D heat equation, J Comput Appl Math 145 (2002) 459–482 J Cheng, J.J Liu, A quasi Tikhonov regularization for a two-dimensional backward heat problem by a fundamental solution, Inverse Probl 24 (2008) 065012 Z Qian, C.L Fu, Regularization strategies for a two-dimensional inverse heat conduction problem, Inverse Probl 23 (2007) 1053 T Regin´ska, K Regins´ki, Approximate solution of a Cauchy problem for the Helmholtz equation, Inverse Probl 22 (2006) 975–989 R.E Showalter, Cauchy Problem for Hyper-parabolic Partial Differential Equations, Trends in the Theory and Practice of Non-Linear Analysis, Elsevier, 1983 A.N Tikhonov, V.Y Arsenin, Solutions of Ill-posed Problems, Winston, Washington, 1977 D.D Trong, N.H Tuan, Regularization and error estimates for nonhomogeneous backward heat problems, Electron J Diff Eqns 04 (2006) 1–10 D.D Trong, N.H Tuan, A nonhomogeneous backward heat problem: regularization and error estimates, Electron J Diff Eqns 33 (2008) 1–14 D.D Trong, P.H Quan, T.V Khanh, N.H Tuan, A nonlinear case of the 1-D backward heat problem: regularization and error estimate, Z Anal Anwend 26 (2) (2007) 231–245 D.D Trong, N.H Tuan, Remarks on a 2-D nonlinear backward heat problem using a truncated Fourier series method, Electron J Diff Eqns 77 (2009) 1– 13 D.D Trong, N.H Tuan, A new regularized method for two dimensional nonhomogeneous backward heat problem, Appl Math Comput 215 (3) (2009) 873–880 ... from financial mathematics, where the celebrated Black–Scholes model [2] for call option can be transformed into a backward parabolic equation whose form is related closely to backward heat equations... some numerical experiments for our method For simplicity, we shall recover the initial temperature at t = from the final data at T = 3431 P.T Nam et al / Applied Mathematics and Computation 216... estimates are available in the next section An essential point here is that the regularized solution is the same in any case This is a substantial pleasure for practical application because even