DSpace at VNU: A modified integral equation method of the semilinear backward heat problem

Applied Mathematics and Computation 217 (2011) 5177–5185 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc A modified integral equation method of the semilinear backward heat problem Nguyen Huy Tuan a,⇑, Dang Duc Trong b, Pham Hoang Quan a a b Department of Mathematics and Application, Saigon University, 273 An Duong Vuong, Ho Chi Minh City, Viet Nam Department of Mathematics, University of Natural Science, Vietnam National University, 227 Nguyen Van Cu, Q5, Ho Chi Minh City, Viet Nam a r t i c l e i n f o Keywords: Backward heat problem Ill-posed problem Nonlinear heat Contraction principle a b s t r a c t The paper is concerned with the non-linear backward heat equation in the rectangle domain The problem is severely ill-posed We shall use a modified integral equation method to regularize the nonlinear problem The error estimates of Hölder type of the regularized solutions are obtained Numerical results are presented to illustrate the accuracy and efficiency of the method This work is a generalization of many earlier papers, including the recent paper [D.D Trong, N.H Tuan, Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation, Nonlinear Anal 71 (9) (2009) 4167–4176] Ó 2010 Elsevier Inc All rights reserved Introduction The backward heat conduction problem (BHCP) is that of finding the distribution of temperature from the final (time) data The problem is ill-posed in the sense of Hadamard In fact, for a given final data, we are not sure that a solution of the problem exists In case a solution exists, it may not depend continuously on the final data This is a typical example of the inverse and ill-posed problems For its applications we refer to various excellent literature, e.g Lattes and Lions [14] and Tikhonov and Arsenin [21] To find approximate solutions for this problem, many approaches have been investigated Lattes and Lions [14], Showalter [20], Ames and Hughes [1] and Miller [17] used quasi-reversibility method Schroter and Tautenhahn [29] established an optimal error estimate for a special BHCP A mollification method has been studied by Hao in [12] Kirkup and Wadsworth used an operator-splitting method in [16] This problem was also investigated by many other authors Dokuchaev [4], Engl et al [5], Hassanov and Mueller [13], Lesnic et al [15], and Yildiz et al [30,31] Although there are many works on the homogeneous case and the linear inhomogeneous cases, the literature on the nonlinear case of the backward heat problem is quite scarce In 2005, Quan and Dung [18] offered a regularized solution by semi-group method However, they gave an error estimate only in a very specific case in which the exact solution has a finite Fourier series expansion In 2007, Trong et al [23] used the quasi-boundary value method to treat the nonlinear case t and attained an error estimate of order T for each t > This estimate is good at any fixed t > but useless at t ¼ Very t t recently, Trong and Tuan [26] improved this method to give an error estimate of order T ln1=ịịT for all t ẵ0; TŠ For the literature on nonhomogeneous and nonlinear backward heat, we refer the reader to the results in Fu et al [8], Trong and his group [19,22–27] However, the error estimate in the mentioned papers is still of logarithmic order In practice, we get the data u by measuring at discrete nodes Hence, instead of u, we shall get an inexact data u satisfying ⇑ 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.2010.11.057 5178 N.H Tuan et al / Applied Mathematics and Computation 217 (2011) 5177–5185 ku À uk ; where the constant  > represents a bound on the measurement error, k Á k denotes the L2-norm The major object of this paper is to provide a new regularization method to estimate the Hölder estimates on [0, 1] We prove that under some suitable conditions, the approximate solution v and the exact solution u satisfy the estimate kuðÁ; tÞ À v  ðÁ; tÞk C m ; m > 0: ð1Þ where C is a constant dependent on u and m is a constant independent of t, u The rest of the paper is organized as follows In the next section, we shall state nonlinear BHCP Then, we review the regularization method and give some estimates In Section 3, we prove the main results Finally, in Section 4, numerical examples are tested to verify the efficacy of the our method Mathematical problem and regularization 2.1 The inverse problem Let u = u(x, t) be the distribution of temperature on the interval ð0; pÞ at the time t and let f ðx; t; uÞ be the heat source which may not be linear in u In fact, we assume that f L1 ð½0; p ẵ0; Rị and that jf x; y; wÞ À f ðx; y; v Þj kjw À v j; where k > independent of x, t, u From the theory of heat conduction, one has the equation ut À uxx ¼ f ðx; t; uðx; tÞÞ; ðx; tÞ ð0; pÞ Â ð0; 1Þ; ð2Þ subject to the boundary condition u0; tị ẳ up; tị ¼ 0; t ð0; 1Þ: The inverse problem is to determine the distribution uðx; tÞ from the final data ux; Tị ẳ uxị: 3ị If a solution exists, then it is the unique solution to the problems (2) and (3) ([23], Theorem 3.1, p 239) For systems (2) and (3), there is no guarantee that the solution exists In the simplest case f = 0, the problems (2) and (3) has a unique solution if and only if X e2n u2n < 1; n¼1 Rp where un ¼ p2 uðxÞ sinðnxÞdx (see [2]) If f ¼ f ðx; tÞ, (see [28, p.43, Lemma 1]) then the problems (2) and (3) has a unique solution if and only if Z  X e n un À 2 esn fn ðsÞds < 1; nẳ1 Rp where fn sị ẳ p2 f x; sị sinnxịdx When f ẳ f x; t; uị, we not know any general condition under which the problems (2) and (3) is solvable In [23], we presented a simple way to check the existence of solution to the systems (2) and (3) (see Theorem 3.2a, p 239) The main purpose of this paper is to find a stable computation method to approximate the exact solution when it exists Recently, we studied this problem in some previous work, for example [26] However, the error estimates in [26] is of logarithmic order, which is not good enough (see [26, Theorem 3, p 4171]) This is a disadvantage point of that paper Here we improve the results in [26,28] by a new regularization method The main idea is to transform the problem into a new form 2.2 Regularization As well known, problems (2) and (3) can be transformed to the following integral equation (see [6]) ux; tị ẳ X et1ịn  un nẳ1 where uxị ẳ X nẳ1 un sinnxị; Z t  e1sịn fn ðuÞðsÞds sin nx; ð4Þ N.H Tuan et al / Applied Mathematics and Computation 217 (2011) 51775185 f uịx; tị ẳ X 5179 fn uịtị sinnxị nẳ1 are the expansions of u and f(u), respectively The term eÀðtÀ1Þn is the unstability cause Hence, in order to regularize the  tÀ1 p where p is problem, we have to replace this term by the better one Naturally, we shall replace this term by  ỵ epn a real number, p P Thus, we shall approximate the problem (4) by the following problem u x; tị ẳ  X  ỵ epn  t1 p un Z nẳ1 where  es1ịn fn u ịsịds sinðnxÞ; t 1: ð5Þ t  is a positive parameter and fn uịtị ẳ p hf x; t; ux; tịị; sinnxịi ẳ 2 un ẳ huxị; sinnxịi ẳ p p Z p Z p p o f ðx; t; uðx; tÞÞ sinðnxÞdx; ð6Þ uðxÞ sinðnxÞdx ð7Þ and hÁ; Ái is the inner product in L2 0; pị If p ẳ then the approximation problem has been studied in [26] We denote W ẳ Cẵ0; 1; L2 0; pịị \ L2 0; 1; H10 ð0; pÞÞ \ C ð0; 1; H10 ð0; pÞÞ The main result of paper is Theorem Let ; M > 0; u L2 ð0; pÞ and let u L2 ð0; pÞ be a measured data such that ku À uk : Then the problem w x; tị ẳ  X  ỵ eÀpn  tÀ1 p un À n¼1 Z  eðsÀ1Þn fn ðw ÞðsÞds sinðnxÞ; t 1; t has a unique solution w W Moreover, if problem (4) has a unique solution u W satisfying 2p sup X e2t1ỵpịn jun tịj2 < M 8ị 06t61 nẳ1 with un tị ẳ p2 Rp uðx; tÞ sin nxdx, then kw ðÁ; tÞ u; tịk p p t1ỵp M ỵ ek ð1ÀtÞ  p : ð9Þ Remark Clark and Oppenheimer [2] considered the following assumptions on the exact solution kuðÁ; 0Þk M : ð10Þ t Under this very weak condition (10) they obtained an error estimate of order T Here, we give a comparison between our results with the results in [2] Note that when p = and f = 0, then (8) becomes (10) v s uX X u 2t2ỵ2pịn2 2 t e un tị ẳ e2tn u2n tị ẳ ku; 0ịk M: pẳ1 nẳ1 Moreover, the error estimate is then of order t , which is the same as that in [2] When p > the condition (8) is very strict However, the error estimate (9) is then of order pÀ1=p This error estimate is much better than the logarithmic order estimates obtained in most of previously known results In work in progress, we are considering the possibility of getting similar estimates like that of (9) under less strict conditions than that of (8) When t ẳ 0, we get ku; 0ị w ; 0ịk p p p1 M ỵ ek  p : The rate of convergence at t ¼ is method  pÀ1 p Hence, for p is large, the term  pÀ1 p can approach  This is a strong point of our 5180 N.H Tuan et al / Applied Mathematics and Computation 217 (2011) 5177–5185  pÀ1  pÀ1 The error (9) is of order O  p for all t ½0; 1Š As we know, the convergence rate of  p ; ðp > 1Þ is faster than that of the À À ÁÁÀq logarithmic order ln 1 ðq > 0Þ when  ! In most of known results, the error between the exact solution and the À ÁÀm regularized solution is of the logarithmic order ln 1 , where m > This type of order is also investigated in many recent papers, such as [2,3,7–11,19,22–27] Combining the above information, the reader can see that our method is effective and useful Proof of the main result First we prove some useful results Lemma For t s 6 p and ðaÞ eðsÀ1Þn ðbÞ   ỵ epn   ỵ epn tÀ1 p tÀ1 p  > 0, we have tÀs 6p; tÀ1 6p: Proof of Lemma Proof a We have es1ịn   ỵ epn t1 p ẳ   s1 p 2 es1ịn  ỵ epn  ts p 2 s1 ẳ  ỵ epn ỵ epn ị p  ts p  ỵ epn  ỵ epn ts p 11ị ts 6p: 12ị Proof b Let s ẳ in Lemma 1a, we get the Lemma 1b h Lemma For all x > 0; < a < we have x ỵ 1ịa x1a : Proof of Lemma Since ð13Þ > 1, and xa < x ỵ 1ịa , we obtain a 1 ỵ x ỵ x1a x ỵ 1ịa ẵ1 ỵ x1a x ỵ 1ịa a : This implies x ỵ 1ịa ẳ x ỵ 1ịa 1 ỵ x1a x ỵ 1ịa ẳ x1a : x ỵ 1ịa x ỵ 1Þa à Lemma Let u L2 ð0; pÞ Then problem (5) has a unique weak solution u ðx; tị W Proof For w Cẵ0; 1; L2 0; pịị, dene Gwịx; tị ẳ /x; tị Z X nẳ1 es1ịn   ỵ eÀpn tÀ1 p  fn ðwÞðsÞds sinðnxÞ t and /ðx; tị ẳ  X  ỵ epn t1 p un sinnxị: nẳ1 We claim that, for every w; v Cẵ0; 1; L2 0; pịị; m P 1, we obtain kGm ðwÞðÁ; tÞ À Gm ðv ÞðÁ; tÞk2  2m k ð1 À tÞm jjjw À v jjj2 ;  m! where jjj Á jjj is the sup norm in Cẵ0; 1; L2 0; pịị We shall prove the latter inequality by induction ð14Þ 5181 N.H Tuan et al / Applied Mathematics and Computation 217 (2011) 5177–5185 For m ¼ 1, using Lemmas and 2, we have kGwị; tị Gv ị; tịk2 ẳ 6 ¼ Z pX p 2 Z k   ỵ epn t1 p !2 ðfn ðwÞðsÞ À fn ðv ÞðsÞÞds  2 Z  t1 p 2 es1ịn  ỵ epn ds t n¼1 X n¼1  2 ðfn ðwÞðsÞ À fn ðv ÞðsÞÞ ds t ð1 À tÞ Z p ð1 À tÞ 2 fn wịsị fn v ịsịị ds ẳ t Z ð1 À tÞ Z p t Z  es1ịn t nẳ1 pX ð1 À tÞ Z t 1 X ðfn wịsị fn v ịsịị ds nẳ1 f x; s; wðx; sÞÞ À f ðx; s; v ðx; sÞÞÞ dxds Z p jwðx; sÞ À v x; sịj2 dxds ẳ C t k  ð1 À tÞjjjw À v jjj2 : ð15Þ Thus (14) holds for m ¼ Suppose that (14) holds for m ¼ j We prove that (14) holds for m ẳ j ỵ Using Lemmas and again, we have kGjỵ1 wị; tị Gjỵ1 v ị; tịk2 ¼ pX Z eðsÀ1Þn Z X 6   ỵ epn   !2 tÀ1 p fn ðGj ðwÞÞðsÞ À fn ðGj ðv ÞÞðsÞ ds !2 jfn ðGj ðwÞÞðsÞ À fn Gj v ịịsịjds t nẳ1 p tị 2  t nẳ1 p 2 ð1 À tÞ Z t ð1 À tÞk 2  ð1 À tÞk Z t 1 X jfn ðGj ðwÞÞðsÞ À fn Gj v ịịsịj2 ds nẳ1 kf ; s; Gj ðwÞðÁ; sÞÞ À f ðÁ; s; Gj ðv ÞðÁ; sÞÞk2 ds Z kGj ðwÞðÁ; sÞ À Gj ðv ÞðÁ; sÞk2 ds t   2j Z k  t  2jỵ1ị sịj k tịjỵ1 j dsC jjjw v jjj2 jjjw v jjj2 : j!  j ỵ 1ị! 16ị Therefore, by the induction principle, we have jjjGm ðwÞ À Gm ðv Þjjj  m k pffiffiffiffiffiffi jjjw À v jjj  m! for all w; v Cð½0; 1; L2 0; pịị We consider G : Cẵ0; 1; L2 0; pịị ! Cẵ0; 1; L2 0; pịị Since  m k pffiffiffiffiffiffi ¼ 0; m!1  m! lim there exists a positive integer number m0 such that Gm0 is a contraction It follows that the equation Gm0 wị ẳ w has a unique solution u Cẵ0; 1; L2 0; pịị We claim that Gu ị ẳ u In fact, one has GGm0 u ịị ẳ Gu ị Hence Gm0 Gu ịị ẳ Gu ị By the uniqueness of the fixed point of Gm0 , one has Gu ị ẳ u , i.e., the equation Gwị ¼ w has a unique solution u Cð½0; 1Š; L2 ð0; pÞÞ h Lemma The solution of the problem (5) depends continuously on u in L2 ð0; pÞ Let u and v be two solutions of (5) corresponding to the final values u and x, respectively From (5) one has in view of the inequality a ỵ bị2 2a2 ỵ b ị ku; tị v ; tịk2 ẳ   t1 pX   þ eÀpn2 p ðu À xn Þ À n  6p Z t1 p  ỵ epn es1ịn  t1 p   ỵ epn t n¼1  X 2 Z X jun xn j ỵ p nẳ1 nẳ1 es1ịn 2  fn uịsị fn v ịsịdsị  ỵ eÀpn t From Lemmas and 2, we get kuðÁ; tÞ À v ðÁ; tÞk2 2 2tÀ2 p 2t ku xk2 ỵ 2k tị p Z À t 2s p kuðÁ; sÞ À v ðÁ; sÞk2 ds: tÀ1 p jfn ðuÞðsÞ À fn ðv ÞðsÞjdsÞ 2 : 5182 N.H Tuan et al / Applied Mathematics and Computation 217 (2011) 5177–5185 Hence À2t p  À2 Z kuðÁ; tÞ À v ðÁ; tÞk2 2 p ku xk2 ỵ 2k tÞ 2s p À kuðÁ; sÞ À v ðÁ; sÞk2 ds: t Using Gronwall’s inequality we have kuðÁ; tÞ À v ðÁ; tÞk pffiffiffi tÀ1 2 p expfk ð1 À tÞ2 gku À xk: Proof of Theorem Let v  be the solution of problem (5) corresponding to u and let w be the solution of problem (5) corresponding to u Using the triangle inequality, we get kw ðÁ; tÞ À uðÁ; tÞk kw ; tị v  ; tịk ỵ kv  ðÁ; tÞ À uðÁ; tÞk: ð17Þ We divide the proof into two steps Step Estimate kw ðÁ; tÞ À v  ðÁ; tÞk Using Lemma 2, we get kw ðÁ; tÞ À uðÁ; tÞk kw ðÁ; tÞ À v  ; tịk p t1 p t1ỵp 2 p expðk ð1 À tÞ2 Þku À u k 2ek ð1ÀtÞ  p : ð18Þ Step Estimate ku ðÁ; tÞ À v  ðÁ; tÞk Suppose the problems (2) and (3) has a unique solution u, we get the following formula ux; tị ẳ Z  X eÀðtÀ1Þn un À  eÀðtÀsÞn fn uịsịds sinnxị: 19ị t nẳ1 Hence Z un tị ẳ et1ịn un etsịn fn uịsịds: 20ị t Multiplying (20) by ỵ epn ị tÀ1 p we have: Z  tÀ1  tÀ1 p p 2 ỵ epn un x; tị ẳ ỵ epn et1ịn un Z 1 t1 p ỵ epn 1 ỵ epn t1 p t 2 etsịn fn uịsịds ẳ es1ịn e1tịn fn uịsịds ẳ  t1 p  ỵ epn t un   ỵ en Z t1 es1ịn  un  ỵ eÀpn tÀ1 p fn ðuÞðsÞds: t ð21Þ From (5) we have: un tị ẳ   ỵ epn tÀ1 p un À Z eðsÀ1Þn  2  ỵ epn t1 p fn u ịsịds: 22ị t Because of (21) and (22), using Lemmas and 2, we have: tÀ1 tÀ1 jun ðtÞ À un tịj jun tị ỵ epn ị p un tịj ỵ ỵ epn ị p Þjun ðtÞj Z eðsÀ1Þn   þ eÀpn tÀ1 p jfn ðu ÞðsÞ À fn ðuÞðsÞjds t t1ỵp p et1ỵpịn jun tịj ỵ Applying the inequality a ỵ bị2 2a2 ỵ b ị and Lemma 1, we obtain kuðÁ; tÞ À u ðÁ; tịk2 ẳ pX 6p jun tị un tịj2 nẳ1 Z X Z X 2s1ịn2 e t1 p 2 X 2t2ỵ2p jfn u ịsị fn uịsịds ỵ 2 p p e2t1ỵpịn jun tịj2  ỵe pn2 2t1 p jfn u ịsị fn uịsịj2 ds ỵ 2 2t2ỵ2p p Z 2ts p  t p X e2t1ỵpịn jun tịj2 nẳ1 t nẳ1 2t  ỵ epn nẳ1 1 Z X 2 p  t n¼1 6p es1ịn t nẳ1 6p 2t2ỵ2p p jfn u ịsị fn uịsịj2 ds ỵ 2 p X 2t1ỵpịn2 e jun tịj2 nẳ1 2s p  kf ; s; uðÁ; sÞÞ À f ðÁ; s; u ðÁ; sÞÞk2 ds ỵ 2p 2t2ỵ2p p X nẳ1 e2t1ỵpịn jun ðtÞj2 ð23Þ 5183 N.H Tuan et al / Applied Mathematics and Computation 217 (2011) 5177–5185 Hence 2t kuðÁ; tÞ À u ðÁ; tÞk2 2 p Z 2s p  2t2ỵ2p p kf ; s; u; sịị f ; s; u ; sịịk2 ds ỵ 2 p t 2t p kuðÁ; tÞ À u ðÁ; tÞk2  2ỵ2p p e2t1ỵpịn jun tịj2 : nẳ1 This implies  X M ỵ 2k Z 2s p À kuðÁ; sÞ À u ðÁ; sÞk2 ds: t Using Gronwall’s inequality, we get À 2t p kuðÁ; tÞ À u ðÁ; tÞk2  It follows that ku; tị u ; tịk 2ỵ2p p Me2k 1tị : p t1ỵp Mek 1tị  p : From (17), (18) and (24), we obtain kw ðÁ; tÞ À uðÁ; tÞk kw ðÁ; tÞ À v  ; tị ỵ kv  ; tị u; tịk p p t1ỵp M ỵ ek ð1ÀtÞ  p ; ð24Þ pffiffiffi tÀ1 pffiffiffiffiffi t1ỵp 2 p expk tị2 ịku u k ỵ Mek 1tị  p for every t ½0; 1Š This completes the proof of Theorem h Numerical experiment In this section, we will describe a numerical implementation of problems (2) and (3) Example (The linear case) Let us consider the linear backward heat problem uxx ỵ ut ẳ ux; tị ỵ gx; tị; u0; tị ẳ up; tị ẳ 0; ux; 1ị ẳ uðxÞ; ðx; tÞ ð0; pÞ Â ð0; 1Þ; t ẵ0; 1; x ẵ0; p; where gx; tị ¼ et sin x and uðx; 1Þ ¼ u0 ðxÞ  e sin x: The exact solution of the equation is ux; tị ẳ et sin x: Especially     999 999  uxị ẳ exp sin x % 2:715564905 sin x: u x; 100 1000 Let u xị  uxị ẳ  ỵ 1ịe sin x We have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi Z p p 2 ku À uk2 ¼  e sin xdx ¼ e : 999 Applying the method introduced in this paper, we find the regularized solution u ðx; 1000 Þ  u xị having the following form u xị ẳ v m xị ẳ w1;m sin x ỵ w6;m sin 6x; where v xị ẳ  ỵ 1ịe sin x; w1;1 ẳ  ỵ 1ịe; w6;1 ẳ 0; Let p ¼ and a ¼ 5000 ; > > < t m ¼ À am m ¼ 1; 2; ; 5; t Àtm  > Á  R tm ðsÀt Þi2 ÀR p > : wi;mỵ1 ẳ  ỵ e2i2 ị mỵ12 wi;m p2 tmỵ1 e m v m xị ỵ gx; sịị sin ix dx ds ; i ¼ 1; 6: 5184 N.H Tuan et al / Applied Mathematics and Computation 217 (2011) 5177–5185 Let a be the error between the regularization solution u and the exact solution u, i.e., a ¼ ku À uk and  ¼ 1 ¼ 10À3 ;  ¼ 2 ¼ 10À7 ;  ¼ 3 ¼ 10À11 , we lead to the first table  À3 1 ¼ 10 2 ¼ 10À4 3 ¼ 10À11 u a 2:718275536 sinðxÞ À 0:005455669367 sinð6xÞ 0.002740395328 2:715835736 sinðxÞ À 0:005459510466 sinð6xÞ 0.0003006372545 2:715562882 sinðxÞ À 0:005504563418 sinð6xÞ 0.00003232321842 Example (The nonlinear case) Let us consider the nonlinear backward heat problem ðx; tÞ ð0; pÞ Â ð0; 1ị; uxx ỵ ut ẳ f uị ỵ gx; tị; u0; tị ẳ up; tị ẳ 0; ux; 1ị ẳ uxị; where t ẵ0; 1; x ẵ0; p; u > > > e10 < e21 À e1 u ỵ e1 f uị ẳ 10 21 e e > u ỵ e1 > > : e1 u ½Àe10 ; e10 Š; u ðe10 ; e11 Š; u ðÀe11 ; Àe10 Š; juj > e11 ; gx; tị ẳ 2et sin x e2t sin x; and ux; 1ị ẳ u0 xị  e sin x: The exact solution of the equation is ux; tị ẳ et sin x: In particular     999 999  uxị ẳ exp sin x % 2:715564905 sin x: u x; 100 1000 Let u xị  uxị ẳ  ỵ 1ịe sin x We have sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi Z p p ku À uk2 ¼ 2 e2 sin2 xdx ¼ e : À 999 Á Applying the method introduced in this paper, we find the regularized solution u x; 1000  u ðxÞ having the following form u xị ẳ v m xị ẳ w1;m sin x ỵ w6;m sin 6x; where v xị ẳ  ỵ 1ịe sin x; w1;1 ẳ  ỵ 1ịe; w6;1 ẳ and a ¼ 5000 > > < tm ¼ À am; m ¼ 1; 2; ; 5; t Àtm  > Á Á  R tm ðsÀt Þi2 R p > : wi;mỵ1 ẳ  ỵ e2i2 ị mỵ12 wi;m p2 tmỵ1 e m v m xị ỵ gx; sị sin ix dx ds ; Letting i ¼ 1; 6:  ¼ 1 ¼ 10À3 ;  ¼ 2 ¼ 10À4 ;  ¼ 3 ¼ 10À11 , we have the table  À3 1 ¼ 10 2 ¼ 10À4 3 ¼ 10À11 u a 2:718264487 sinðxÞ À 0:005466473792 sinð6xÞ 0.002729464336 2:715833791 sinðxÞ À 0:005461493459 sinð6xÞ 0.0002987139108 2:715552177 sinðxÞ À 0:005518178192 sinð6xÞ 0.00004317829056 N.H Tuan et al / Applied Mathematics and Computation 217 (2011) 5177–5185 5185 Acknowledgments This project was supported by National Foundation for Science and Technology Development (NAFOSTED), Code: 101.012010.10 The authors thank the editor and the referees for their valuable comments leading to the improvement of our manuscript The authors thank Truong Trung Tuyen in the Indiana University and Nguyen Minh Quan in the University 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