Two numerical methods for solving a backward heat conduction problem

Size: px

Start display at page:

Download "Two numerical methods for solving a backward heat conduction problem"

Simon Norton
5 years ago
Views:

1 pplied Mathematics and Computation 79 (6) wwwelseviercom/locate/amc Two numerical methods for solving a backward heat conduction problem Xiang-Tuan Xiong, Chu-Li Fu *, hi Qian epartment of Mathematics, Lanzhou University, Lanzhou 73, People s Republic of China bstract We introduce a central difference method and a quasi-reversibility method for solving a backward heat conduction problem (BHCP) numerically For these two numerical methods, we give the stability analysis Meanwhile, we investigate the roles of regularization parameters in these two methods Numerical results show that our algorithm is effective Ó 5 Elsevier Inc ll rights reserved Keywords: Inverse problem; Backward heat conduction; Central difference; Quasi-reversibility; Regularization Introduction The backward heat conduction problem (BHCP) is also referred to as final boundary value problem In general, no solution which satisfies the heat conduction equation with final data and the boundary conditions exists Even if a solution exists, it will not be continuously dependent on the final data The BHCP is a typical example of an ill-posed problem which is unstable by numerical methods and requires special regularization methods In the context of approximation method for this problem, many approaches have been investigated Such authors as Lattes and Lions [], Showalter [], mes et al [3], Miller [4] have approximated the BHCP by quasi-reversibility methods Schröter and Tautenhahn [5] established an optimal error estimate for a special BHCP Mera and Jourhmane used many numerical methods with regularization techniques to approximate the problem in [6 8], etc mollification method has been studied by Hào in[9] Kirkup and Wadsworth used an operator-splitting method in [] difference approximation method for solving sideways heat equation was provided by Eldénin[] So far in the literature (cf [3,] and the references therein), most of the authors used the eigenfunctions and eigenvalues to reconstruct the solution of the BHCP by many quasi-reversibility methods numerically, However, the eigenfunctions and eigenvalues are in general not available and the labor needed to compute these and the corresponding Fourier coefficients is very onerous In this paper, we use two regularization methods to solve the BHCP in one-dimensional setting numerically, but these methods can be generalized to two-dimensional case * Corresponding author addresses: xiongxt4@stlzueducn (X-T Xiong), fuchuli@lzueducn (C-L Fu) 96-33/$ - see front matter Ó 5 Elsevier Inc ll rights reserved doi:6/jamc54

2 The paper is organized as follows In the forthcoming section, we will present the mathematical problem on a BHCP; in Section 3, we review two regularization methods, one is central difference regularization method, the other is a special quasi-reversibility regularization method; in Section 4, some finite difference schemes are constructed for the inverse problem and the numerical stability analysis is provided; in Section 5, numerical examples are tested to verify the effect of the numerical schemes Mathematical problem The direct problem X-T Xiong et al / pplied Mathematics and Computation 79 (6) We consider the following heat equation: u t ðx; tþ u xx ðx; tþ; p < x < p; < t < T ; uðp; tþ sðtþ; < t < T ; uð p; tþ lðtþ; < t < T ; uðx; Þ f ðxþ; p < x < p ð:þ Solving the equation with given s(t); l(t) and f(x) is called a direct problem From the theory of heat equation, we can see that for s(t); l(t) and f(x) in some function space there exists a unique solution [3] The inverse problem Consider the following problem: u t ðx; tþ u xx ðx; tþ; p < x < p; < t < T ; uðp; tþ sðtþ; < t < T ; uð p; tþ lðtþ; < t < T ; uðx; T ÞgðxÞ; p < x < p The inverse problem is to determine the value of u(x, t) for 6 t < T from the data s(t); l(t) and g(x) If the solution exists, then the problem has a unique solution (cf [4, p 64]) The data g(x) are based on (physical) observations and are not known with complete accuracy, due to the ill-posedness of the BHCP, a small error in the data g(x) can cause an arbitrarily large error in the solution u(x, t) Now we want to reconstruct the temperature distribution u(x, t) for 6 t < T by two different regularization methods 3 Central difference regularization By using the central difference with step length h to approximate the second derivative u xx, we can get the following problem: u t ðx; tþ uðxþh;tþ uðx;tþþuðx h;tþ ; p < x < p; < t < T ; h uðp; tþ sðtþ; < t < T ; ð3:þ uð p; tþ lðtþ; < t < T ; uðx; T ÞgðxÞ; p < x < p Let ~t T t; wðx;~tþ uðx; tþ, we have w ~t w xx ; p < x < p; < ~t < T ð3:þ If w xx at x i is replaced by a second-order central difference, then (3) becomes ð:þ w ~t ðx i ;~tþ h ½wðx i þ h;~tþ wðx i ;~tþþwðx i h;~tþš ð3:3þ

3 37 X-T Xiong et al / pplied Mathematics and Computation 79 (6) Let x i = p +(i )h, for i =,,n + h p ; w n i w i ð~tþ wð p þði Þh;~tÞ Then by the boundary condition of (3) there hold w ð~tþ lð~tþ and w nþ ð~tþ sð~tþ Meanwhile, Eq (33) with initial boundary conditions can be discretized as w ð~tþ w ð~tþ w ð~tþ h h h h h w i ð~tþ B B w i ð~tþ B C B h w n ð~tþ h fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} w ~t n ð~tþ w nþ ð~tþ ðn Þðn Þ fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} n w ðþ gðx Þ w i ðþ gðx i Þ ð3:5þ B C w n ðþ gðx n Þ This is an ordinary differential system and there are many numerical methods such as Euler method, Runge Kutta method for the system But we find that the eigenvalues of are k k 4 sin kp P, k (the positive integer set) Therefore, the numerical method for (34) and (35) is unstable However, via the variable h ðn Þ transformation [5] v i ð~tþ e a~t w i ð~tþ; i ; ; n þ ; ð3:6þ where a > is a contraction factor to be determined such that (33) in terms of v i ð~tþ is changed to ov i ð~tþ o~t h a v i ð~tþ h ½v iþð~tþþv i ð~tþš; ð3:7þ similarly, we have v ð~tþ v ð~tþ e a~t lð~tþ a h h a h v i ð~tþ h h B B C C i ð~tþ B h B a h v n ð~tþ h fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} v n ð~tþ e a~t sð~tþ ~t B ðn Þðn Þ v ðþ gðx Þ v i ðþ gðx i Þ ð3:9þ B C v n ðþ gðx n Þ We choose a b=h P h such that the eigenvalues of matrix B are negative, thus the numerical methods such as Runge Kutta method for (38) and (39) are stable fter v i ð~tþ is obtained numerically, we can obtain w i ð~tþ e a~t v i ð~tþ, furthermore wðih;~tþ uðih; T ~tþ u i ðtþ In the central difference regularization method, the space step length h plays the role of regularization parameter ccording to the general regularization theory, the h should be neither too small nor too large ð3:4þ

4 X-T Xiong et al / pplied Mathematics and Computation 79 (6) By our experience, the parameter h is selected as h ln =, where d is the noise level of the data g(x) d We will see this choice rule is very effective in the subsequent numerical tests 4 Quasi-reversibility regularization method The initial boundary value problem () is replaced by the following problem: u t ðx; tþ u xx ðx; tþþu xxt ðx; tþ; p < x < p; < t < T ; uðp; tþ sðtþ; < t < T ; uð p; tþ lðtþ; < t < T ; uðx; T ÞgðxÞ; p < x < p; ð4:þ where is a small positive parameter For sufficiently small the solution of (4) approximates the solution (if it exists) of () in some sense This is one of well-known quasi-reversibility methods [] For the above mentioned problem, Ewing [] has presented a choice rule of the regularization parameter, ie, ln, d where d denotes the noise level of the data g(x), and the error estimate between the approximate solution and the exact solution is given in L (R)-norm Similarly, we take ~t T t, then problem (4) becomes u ~t ðx;~tþþu xx ðx;~tþ u xx~t ðx;~tþ ; p < x < p; < ~t < T ; uðp;~tþ sð~tþ; < ~t < T ; ð4:þ uð p;~tþ lð~tþ; < ~t < T ; uðx; Þ gðxþ; p < x < p The problem has a unique solution if a solution exists Now we prove it for two-dimensional case Theorem There exists a unique solution (if it exists) for the problem: u ~t ðx;~tþþuðx;~tþ u ~t ðx;~tþ ; ð; T Þ; uðx;~tþ hðx;~tþ; on o ð; T Þ; uðx; Þ gðxþ; in ; where is a bounded subset in R, is the Laplace operator, > ð4:3þ Proof We only need to prove the following problem has the zero solution: w ~t ðx;~tþþwðx;~tþ w ~t ðx;~tþ ; ð; T Þ; wðx;~tþ ; on o ð; T Þ; wðx; Þ ; in Set uð~tþ w ð~tþþjrwð~tþj dx; then duð~tþ ww ~t þ rw rw ~t dx d~t ue to the Green s second formula, we have duð~tþ ww ~t ww ~t dx wðw ~t w ~t Þdx d~t 6 w ð~tþþjrwð~tþj dx uð ~tþ wð wþdx jrwj dx ð4:4þ

5 374 X-T Xiong et al / pplied Mathematics and Computation 79 (6) Therefore, we have uð~tþ 6 uðþe ~t ð4:5þ since u() =, there holds uð~tþ 6 Hence w = h Now we construct the finite difference schemes for solving problem (4), let x i = p +(i )h, i =,,n +,~t j ðj Þs, j =,,m +, where h p ; s T Let n m uj i uðx i ;~t j Þ represent the value of the numerical solution of (4) at the mesh point ðx i ; ~t j Þ, then Eq (4) is discretized as ru jþ iþ þ r þ u jþ i ru jþ i s h þ r u j iþ þ h þ r þ u j i s h þ r u j i ; ð4:6þ where r, i =,,n; j =,,m h s Now we discuss the stability of difference schemes (46) by verifying the Von Neumann condition The propagation factor can be found þ 4 þ r sin rh s h Gðr; sþ þ 4r s sin rh It is easy to verify the fact that the Von Neumann condition jgðr; sþj 6 þ cs holds with c Hence, the numerical algorithm (46) is stable 5 Numerical examples For convenience, we take s(t) = l(t) = in() Example We consider the following direct problem: u t ðx; tþ u xx ðx; tþ; p < x < p; < t < ; uðp; tþ ; < t < ; uð p; tþ ; < t < with the initial condition: p þ x; p 6 x 6 ; uðx; Þ p x; 6 x 6 p Let x = p, x i = p +(i )h, i =,,n; x n + =, t j =(j )s, j =,,m +, the space step length h p and the time step length s, then we solve this problem by an explicit difference scheme in n m the following form: with u jþ i ru j iþ þð rþuj i þ ru j i ; j ; ; m; i ; ; n uðx nþ ; t j Þu j nþ ; j ; ; m þ ; ð5:þ uðx ; t j Þu j ; j ; ; m þ uðx i ; t Þ p þ x i; p 6 x i 6 ; p x i ; 6 x i 6 p; where r s, and it requires r < for numerical stability reasons h The numerical result for g(x) = u(x, T = ) is shown in Fig, where n =, m = 5 Now we solve the inverse problem by the g(x) generated numerically by the direct problem via the two different regularization methods We choose to restore the solution f(x) att = We denote the numerical result ð5:þ

6 X-T Xiong et al / pplied Mathematics and Computation 79 (6) g(x) Fig g(x) computed by (5) of the inverse problem as f % (x) If we introduce random noises e to data g(x), ie, g e (x i )=g(x i )+erand(i), where rand(i) is a random number between [, ], then the total noise d can be measured in the sense of root mean square (RMS) error according to vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u Xn d : t ðg n e ðx i Þ gðx i ÞÞ ð5:3þ i In the computation, we choose the regularization parameters h ln for the central difference regularization method and ln d for the quasi-reversibility method, respectively The numerical results are d shown in Fig (quasi-reversibility), Fig 3(central difference) with d = Example Consider the problem u t u xx ; p < x < p; < t < ; uð p; tþ uðp; tþ ; < t < ; ð5:4þ uðx; Þ e sin x; p < x < p 35 3 numerical solution exact solution Fig f % and f, m = 5, n =, = 45

7 376 X-T Xiong et al / pplied Mathematics and Computation 79 (6) numerical solution exact solution Fig 3 f % and f, b =,m = 5, n =4,h = p/4 8 numerical solution exact solution Fig 4 f % and f, m =, n = 3, = 9 8 numerical solution exact solution Fig 5 f % and f, b =3,m = 5, n =5,h = p/5

8 The exact solution is uðx; tþ e t sin x ð5:5þ The numerical result for the inverse problem is also denoted as f % (x) See Fig 4 (quasi-reversibility) and Fig 5 (central difference), where d = From Examples and, we conclude that the choice rules of the regularization parameters h and are very effective By our numerical experiments, we can see that the accuracy of the numerical results increases with the decreasing T; at the same time, the numerical solutions of the quasi-reversibility method depend on the parameter continuously This accords with the theoretical result (cf [6]) Here, we will not give the numerical results 6 Conclusions In this paper, we discussed two regularization methods for the one-dimensional backward heat conduction problem We presented two algorithms for the inverse problem Numerical results show that these method are effective with two appropriately chosen regularization parameters The algorithm for two-dimensional case is to be considered cknowledgements The work is supported by the NNSF of China (Nos 75 and 5779) and the NSF of Gansu Province of China (No 3S5-5-5) References X-T Xiong et al / pplied Mathematics and Computation 79 (6) [] R Lattes, JL Lions, Methode de Quasi-Reversibility et pplications, unod, Paris, 967 (English Transl: R Bellman, Elsevier, New York, 969) [] RE Showalter, The final value problem for evolution equations, J Math nal ppl 47 (974) [3] K mes, WC Gordon, JF Epperson, SF Oppenhermer, comparison of regularizations for an ill-posed problem, Math Comput 67 (998) [4] K Miller, Stabilized quasireversibility 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 36, Springer-Verlag, Berlin, 973, pp 6 76 [5] T Schröter, U Tautenhahn, On optimal regularization methods for the backward heat equation, nal nw 5 (996) [6] NS Mera, L Elliott, B Ingham, Lesnic, n iterative boundary element method for solving the one dimensional backward heat conduction problem, Int J Heat Mass Transf 44 () [7] M Jourhmane, NS Mera, n iterative algorithm for the backward heat conduction problem based on variable relaxtion factors, Inverse Probl Eng () [8] NS Mera, The method of fundamental solutions for the backward heat conduction problem, Inverse Probl Sci Eng 3 () (5) [9] N Hào, mollification method for ill-posed problems, Numer Math 68 (994) [] SM Kirkup, M Wadsworth, Solution of inverse diffusion problems by operator-splitting methods, ppl Math Model 6 () () 3 8 [] L Eldén, Numerical solution of the sideways heat equation by difference approximation in time, Inverse Probl (995) [] RE Ewing, The approximation of certain parabolic equations backward in time by Sobolev equations, SIM J Math nal 6 (975) [3] JR Cannon, The One-imensional Heat Equation, ddison-wesley Publishing Company, California, 984 [4] LC Evans, Partial ifferential Equations, merican Mathematical Society, Providence, Rhode Island, 998 [5] CS Liu, Group preserving scheme for backward heat conduction problems, Int J Heat Mass Transf 47 (4) [6] K mes, LE Payne, Continuous dependence on modeling for some well-posed perturbations of the backward heat equation, J Inequal ppl 3 (999) 5 64

Initial inverse problem in heat equation with Bessel operator

International Journal of Heat and Mass Transfer 45 (22) 2959 2965 www.elsevier.com/locate/ijhmt Initial inverse problem in heat equation with Bessel operator Khalid Masood *, Salim Messaoudi, F.D. Zaman