Finite Difference Method for the Estimation of a Heat Source Dependent on Time Variable ABSTRACT

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1 Malaysia Joural of Matematical Scieces 6(S): 39-5 () Special Editio of Iteratioal Worsop o Matematical Aalysis (IWOMA) Fiite Differece Metod for te Estimatio of a Heat Source Depedet o Time Variable, Allabere Asyralyev ad Abdulla Said Erdoga Departmet of Matematics, Fati Uiversity, Istabul, Turey Departmet of Matematics, ITTU, Asgabat, Turmeista aasyr@fati.edu.tr ad aserdoga@fati.edu.tr ABSTRACT Well-posedess of differece sceme for te iverse problem of recostructig te rigt side of a parabolic equatio u( t, x) u( t, x) = a( x) σu( t, x) + p( t) q( x) + f ( t, x), t x < x <, < t T, ux( t,) =, ux( t, l) = ψ ( t), t T, u(, x) = ϕ( x), x l, u( t, x ) = ρ( t), x l, t T, were u( t, x ) ad p( t ) are uow fuctios, f ( t, x), q( x), ϕ( x), ψ ( t) ad ρ ( t) are give fuctios, a( x) δ > ad σ > is a sufficietly large umber. umerical metods for estimatio of costat terms of coercive stability estimates are described. MSC: 65, 65M, 65J Keywords: Parabolic iverse problem, well-posedess, differece sceme.. ITRODUCTIO Te iverse problem of recostructig te rigt ad side of a parabolic equatio as bee cosidered i may papers (see Boruov ad Vabiscevic (), Samarsii ad Vabiscevic (7) ad te refereces terei). Te iverse eat coductio problems deal wit te determiatio of te crucial parameters i aalysis suc as determiatio of boudary coditios, te iteral eergy source, termal coductivity, te volumetric eat capacity, etc. Tey ave bee widely applied i may desigs ad maufacturig problems especially i wic direct measuremets of surface

2 Allabere Asyralyev & Abdulla Said Erdoga coditios are ot possible. Te formulatio of umerical metods ad literature review is give by may researcers. I order to determie uow coditios, tese metods ave ofte bee combied wit te optimizatio algoritms suc as regularizatio tecique. Te teoretical statemets o well-posedess of te iverse problem wit oe variable as bee cosidered i may teoretical papers (Ivacov (995), Coulli ad Yamamoto (996, 999) ad Asyralyev ()). Te geeric well-posedess of a liear iverse problem is studied for values of a diffusio parameter ad geeric local well-posedess of a iverse problem is proved i Coulli ad Yamamoto (996, 999) were te uow cotrol fuctio is i space variable. I Boruov ad Vabiscevic () ad Samarsii ad Vabiscevic (7), te well-posedess of te algoritm for te umerical solutio of te idetificatio problem wit time variable is ivestigated i maximum orm. I Asyralyev (), te well-posedess of problem of determiig te parameter of a parabolic equatio is cosidered i Hölder spaces. A omogeous plate wit l ticess ad costat termal properties wit isulated boudaries eated by a plae surface eat source of p( t ) located at a specified positio x = x ca be formulated as a parabolic equatio (Liu (8) ad Yag (998)). Also, i te process of trasportatio, diffusio ad coductio of atural materials, te followig eat equatio is iduced (Ya et al. (8)) ut a u f t x u t x o tmax = (, ; ), (, ) (, ] Ω, were u represets state variable, a is te diffusio coefficiet, Ω is a d bouded domai i R ad f deotes pysical laws, wic meas source terms ere.. FIRST ORDER OF ACCURACY DIFFERECE SCHEMES AD THE WELL-POSEDESS We cosider te iverse problem of recostructig te rigt side of a parabolic equatio wit olocal coditios Malaysia Joural of Matematical Scieces

3 Fiite Differece Metod for te Estimatio of a Heat Source Depedet o Time Variable u( t, x) u( t, x) = a( x) σu( t, x) + p( t) q( x) + f ( t, x), t x < x <, < t T, ux ( t,) =, ux ( t, l) = ψ ( t), t T, u(, x) = ϕ( x), x l, u( t, x ) = ρ( t), x l, t T, () were u( t, x ) ad p( t ) are uow fuctios, f ( t, x), q( x), ϕ( x), ψ ( t) ad ρ ( t) are give fuctios, a( x) δ > ad σ > is a sufficietly large umber. Here x is te iterior locatio of a termocouple recordig te temperature measuremet. Assume tat (a) q( x ) is a sufficietly smoot fuctio, (b) q () = q ( l) =, (c) q ( x ). Te first order of accuracy differece sceme for te approximate solutio of te problem () u u u+ u + u = a( x ) σ u (, ), + p q + f t x p = p( t ), q = q( x ), x =, t =,, M, M = l, = T, u u =, um um = ψ ( t ),, u = ϕ( x ), M, u = u ( ),, x s = ρ t s M [ ] () ere qs, q = q ad qm = qm are assumed is costructed. Malaysia Joural of Matematical Scieces

4 α α Allabere Asyralyev & Abdulla Said Erdoga To formulate our results, we itroduce te Baac space M C = C [, l], α (,) of all grid fuctios φ = { φ } defied o [, l] = { x =, M, M = l} wit φ = φm equipped wit te orm = α φ = max φ + sup φ φ ( r). α C + r M + r M C ( E) = C([ o, T], E) is te Baac space of all grid fuctios { (, )} φ = φ t = defied o [, T] { t = t,, = T} wit values i E equipped wit te orm φ C ( E) = max φ( t ). E Let A be a strogly positive operator. Wit te elp of A we itroduce te fractioal spaces Eα ( E, A), < α <, cosistig of all υ E for wic te followig orms are fiite υ = υ + + υ α E sup α ( ) A A. > E Trougout te article costats are idicated by M ( α, β, ) were te costat depeds oly o α, β,. Te, te followig teorem o wellposedess of problem () is establised. Teorem. For te solutio problem (), te followig coercive stability estimates u u + { D u } α α = C C = C C ρ( t ) ρ( t ) M ( q, s) = C[, T ] Malaysia Joural of Matematical Scieces

5 Fiite Differece Metod for te Estimatio of a Heat Source Depedet o Time Variable + M aɶ T D + f t α + { } (,,, ) α φ α ϕ ( ) ρ, C = C C C[, T ] ( t ) ( t ) M ( q, s) C[, T ] = C[, T ] ρ ρ p (,,, ) + M aɶ φ α T Dϕ α + { f ( t )} α + ρ C = C C C[, T ] old. Here, f t = { f t x } ϕ = { ϕ x } ρ = { ρ t } M M = = = M u+ u + u = ad ( aɶ = ad q σ q ). = qs D u ( ) (, ), ( ), ( ), Te proof of teorem is based o te iequality p M ( q, s) max + max ρ( t ) ρ( t ) w w α C were { } w = is te solutio of te followig differece sceme w w w w + w t w q q + q ρ( t ) ws q σ σ w + f ( t, x ), x =, t =, q + ρ( ) s + = a( x ) a( x ) qs s, M, M = l, = T, w w =, wm wm = ψ ( t ),, w = ϕ( x ), M, (3) ad te followig teorems. Malaysia Joural of Matematical Scieces 3

6 Allabere Asyralyev & Abdulla Said Erdoga Teorem. Te Followig coercive stability estimate w w α = C C { ( )} M ( aɶ, φ, α, T) ρ C C[, T ] ϕ α f t α + = + C C olds. x Teorem 3. For < α < te orms of te spaces Eα ( C[, l], A ) ad α C [, l] are equivalet. 3. UMERICAL RESULTS For te umerical verificatio of our algoritm, we assume tat te diffusio coefficiet a =, q( x) = ad f ( t, x ) =. We cosider te followig problem u( t, x) u( t, x) = + p( t), x (, π ), t (,], t x u(, x) = cos x + x, x [, π ], 3 ux t ux t tπ π t = = + t 3 u t = ρ t = e + t + t 6 (,), (,) 6, [,],, ( ) cos, [,]. () Te exact solutio of te give problem is ad of te cotrol parameter p( t ) is 6 t. t u( t, x) = e cos x + 3tx + x Malaysia Joural of Matematical Scieces

7 Fiite Differece Metod for te Estimatio of a Heat Source Depedet o Time Variable First, applyig te Rote differece sceme (), u u u+ u + u = + p, p = p( t ),, t =, M, M = π, =, u = cos( x ) + ( x ), x = M, um u M 3 u = u, = 6 tπ + π, t =,, us = ρ( t ) = cos e + t +, t =,, s = t 3 6 (5) is costructed. We eed to calculate te approximate value of te cotrol parameter p( t ). Te value of p( t ) at te grid poits ca be obtaied from te equatio ρ( t ) ρ( t ) ws ws p =,, (6) were w r, r =, is te solutio of te differece sceme s w w w + w + w =,, M, M = π, = T, wm wm 3 w = w, = 6 tπ + π, t =,, w = cos( x ) + ( x ), M. (7) Malaysia Joural of Matematical Scieces 5

8 Allabere Asyralyev & Abdulla Said Erdoga Te differece sceme (7) ca be arraged as w w + w + w =,, M, M = l, = T wm wm 3 w = w, = 6 tπ + π, t =,, w = cos( x ) + ( x ), M. First, applyig te first order of accuracy differece sceme (7), we obtai ( + ) ( M + ) system of liear equatios ad we write tem i te matrix form { ( ) ( ) } Aw Bw Dϕ,, w cos x x + = = + (8) M = were. x y x. x y x. A = x y x. M M ( + ) ( + ). v. v. B = M M ( + ) ( + ) 6 Malaysia Joural of Matematical Scieces

9 Fiite Differece Metod for te Estimatio of a Heat Source Depedet o Time Variable Here, x = y = + +, v =, w r, r w = r w M ( M + ) f or r =, ϕ = ad D is ( M ) ( M ) 3 ( 6tπ + π ) ( M ) idetity matrix. Usig (8), we ca obtai tat { } ( ϕ ),,,...,, cos( ) ( ) w A D Bw w x x = = = + (9) Te, we ca reac to te solutio of w,, M. Applyig te first order of accuracy differece sceme (5) ad (6), we ave agai ( + ) ( M + ) system of liear equatios ad we write tem i te matrix form were A + B, cos u u = D u = x, x ϕ + ( ) ( ) M = M = were Here, u r A = A, B = B r u = f or r =,, r u M ( M + ) Malaysia Joural of Matematical Scieces 7

10 Allabere Asyralyev & Abdulla Said Erdoga φ φ =, φ = p φ M 3 ( 6tπ + π ) ( M ) + To solve te resultig differece equatios, we agai apply te iterative metod give i (9). ow, we will give te results of te umerical aalysis. Te umerical solutios are recorded for differet values of ad M ad u represets te umerical solutios of tese differece scemes at (, ) t x. Table gives te relative error betwee te exact solutio of p( t ) ad te solutios derived by te umerical process. Te error is computed by E p max p( t ) p = max p( t ). TABLE : Error aalysis for p( t ). =3 =6 =9 Rel. Error Table gives te error aalysis betwee te exact solutio ad te solutios derived by differece scemes. Table costructed for = M = 3,6 ad 9 respectively. For teir compariso, te error is computed by E = max u( t, x ) u M max u( t, x ) M. 8 Malaysia Joural of Matematical Scieces

11 Fiite Differece Metod for te Estimatio of a Heat Source Depedet o Time Variable TABLE : Error aalysis for te exact solutio u( t, x ). Metod =M=3 =M=6 =M=9 st order accuracy d.s...3 Te obtaied results also sow tat te umerical solutios are stable ad coverge to te exact solutio. A similar approac ca be applied to geeral boudary coditios. Hig order accuracy differece sceme ca be ivestigated by usig te operator approaces. REFERECES Asyralyev, A.. O a problem of determiig te parameter of a parabolic equatio. Ur. Mat. J. 9: -. Boruov, V. T. ad Vabiscevic, P... umerical solutio of te iverse problem of recostructig a distributed rigt-ad side of a parabolic equatio. Comput. Pys. Commu. 6: Cao, J. R. ad Yi, Hog-Mig. 99. umerical solutios of some parabolic iverse problems. umer. Met. Part. D. E. : Cao, J. R., Li, Y. ad Wag, S. 99. Determiatio of a cotrol parameter i a parabolic differetial equatio. J. Austral. Mat. Soc., Ser. B. 33: Coulli, M. ad Yamamoto, M Geeric well-posedess of a iverse parabolic problem-te Hölder-space approac. Iverse Probl. : Coulli, M. ad Yamamoto, M Geeric well-posedess of a liear iverse parabolic problem wit diffusio parameter. J.Iv. Ill- Posed Problems. 7(3): -5. Dega, M. 3. Fidig a cotrol parameter i oe-dimesioal parabolic equatios. Appl. Mat. Comput. 35: Ivacov,. I O te determiatio of uow source i te eat equatio wit olocal boudary coditios. Ur. Mat. J. 7(): 38-. Malaysia Joural of Matematical Scieces 9

12 Allabere Asyralyev & Abdulla Said Erdoga Liu, Fug-Bao. 8. A modified geetic algoritm for solvig te iverse eat trasfer problem of estimatig pla eat source. It. J. Heat Mass Tra. 5: Prilepo, A. I. ad Kosti, A. B. 99. Some iverse problems for parabolic equatios wit fial ad itegral observatio. Mat. Sb. 83(): Samarsii, A. A. ad Vabiscevic, P.. 7. umerical metods for solvig iverse problems of matematical pysics. Iverse ad Illposed Problems Series. Berli, ewyor: Walter de Gruyter. Yag, Cig-Yu A sequetial metod to estimate te stregt of te eat source based o symbolic computatio. It. J. Heat Mass Tra. (): 5-5. Ya, L., Fu, Cu-Li ad Yag, Feg-Lia. 8. Te metod of fudametal solutios for te iverse eat source problem. Eg. Aal. Boud. Elem. 3: Malaysia Joural of Matematical Scieces