LAPLACE TRANSFORM SOLUTION OF THE PROBLEM OF TIME-FRACTIONAL HEAT CONDUCTION IN A TWO-LAYERED SLAB

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1 Journl of Appled Mthemtcs nd Computtonl Mechncs 5, 4(4), p-issn DOI:.75/jmcm.5.4. e-issn LAPLACE TRANSFORM SOLUTION OF THE PROBLEM OF TIME-FRACTIONAL HEAT CONDUCTION IN A TWO-LAYERED SLAB Stnsłw Kul, Urszul Sedlec Insttute of Mthemtcs, Czestochow Unversty of Technology Częstochow, Polnd stnslw.ul@m.pcz.pl, urszul.sedlec@m.pcz.pl Abstrct. In ths pper the Lplce trnsformton for solvng the problem of frctonl het conducton n two-lyered slb hs been ppled. The dfferent orders of Cputo dervtve n the tme-frctonl equton governed the het trnsfer n the lyers re ssumed. The nverse Lplce trnsform by usng numercl method s determned. The numercl results obtned by usng of the egenfunctons method nd by numerclly nvertng the Lplce trnsform re compred. Keywords: frctonl het conducton, two-lyered slb, Lplce trnsformton. Introducton Clsscl models of the het conducton re derved wth ssumpton of the Fourer lw of het trnsfer. The het conducton problems n mult-lyered bodes bsed on the Fourer lw s the subject of pper [] by Hj-Sheh nd Bec. Exct solutons of these problems for mult-lyered slbs, cylnders nd spheres re presented by Özş n boo []. A generlzton of the Fourer lw leds to frctonl het conducton. The frctonl het equton ncludes frctonl dervtve wth respect to spce nd/or tme vrble. The problems of frctonl het conducton re the subject of numerous wors, for nstnce references [3-7]. In boo [3] by Povsteno the equtons obtned by generlztons of the tme-nonlocl Fourer s, Fc s nd Drcy s lws re dscussed. The equtons n Crtesn, polr, cylndrcl nd sphercl coordntes re consdered. The problems of het conducton n semnfnte or n nfnte composte medum consstng of two regons chrcterzng by dfferent orders of the tme-frctonl Cputo dervtve n the het equton were studed by Povsteno n ppers [4-7]. The presented solutons re derved wth ssumpton tht the two consdered solds re n perfect therml contct. The het conducton n mult-lyer slb governed by tme-frctonl equton s dscussed by Sedlec nd Kul n reference [8]. The clsscl convectve bound-

2 6 S. Kul, U. Sedlec ry condtons nd the clsscl condtons descrbng the perfect contct of the solds were ssumed. To solvng the problem the egenfunctons method ws ppled. Applcton of the Lplce trnsformton to solvng the frctonl dfferentl equton s shown by Podlubny n the boo [9]. The nverse of the Lplce trnsform cn be determned numerclly. The methods for numercl nverson of the Lplce trnsforms re presented n ppers [, ]. In ths pper, soluton of the tme-frctonl het conducton n two-lyered slb s presented. The physcl Robn condton on the sphere surfce nd the perfect contct of the lyers re ssumed. The Lplce trnsformton s ppled nd the temperture n the slb s obtned by usng method of numercl nverson of the Lplce trnsforms.. Formulton of the problem Consder slb consstng of two lyers wth therml conductvty λ nd therml dffusvty. The tme-frctonl dfferentl equton of the het conducton n the -th lyer s T T + q ( xt, ) =, x [ x, x ], <, =, λ t where T ( xt, ) s temperture n the -th lyer nd (, ) () q xt s volumetrc energy generton, x =, x nd x re coordntes specfyng surfces of the slb boundres nd n nterfce between the lyers, respectvely, denotes n order of the Cputo frctonl dervtve wth respect to tme t. The Cputo dervtve of order s defned by [9] C ( m ) t ( τ) d f t m d f Dt f ( t) = = ( t τ) dτ, m < < m m dt Γ () dτ We ssume the Robn boundry condtons [3] t x= nd x= x m λ T ( x, t) = T ( t) T ( x, t) DRL L L (3) T λ = ( x, t) R( TR( t) T ( x, t) ) DRL where L, R re het trnsfer coeffcents, L, TR t re surroundng tempertures nd D RL denotes the Remnn-Louvlle frctonl dervtve of order. Moreover, the condtons of the perfect contct of the lyers re stsfed [3] T t (4)

3 Lplce trnsform soluton of the problem of tme-frctonl het conducton n two-lyered slb 7 (, ) (, ) T x t = T x t (5) T T λdrl x t DRL x t (, ) = λ (, ) (6) nd the ntl condton s [ ] T x, = f x, x x, x, =, (7) The Remnn-Louvlle frctonl dervtve occurrng n equtons (3), (4) nd (6) s defned by [9] d m DRLf ( t) = ( t τ) f ( τ) dτ, m < < m Γ m dt (8) m t m The condtons (3),(4) of the convectve het trnsfer between the slb nd the surroundngs re clled the physcl Robn condtons [3]. Substtutng = = n equtons (3),(4) the clsscl condtons of the thrd nd re obtned. These condtons n the theory of frctonl het trnsfer re lso clled mthemtcl Robn condtons [3]. In ths cse the condton (6) of equlty of the het fluxes t the nterfce ssumes lso clsscl form. A soluton of the het conducton problem n the slb under the physcl Robn boundry condtons by usng the Lplce trnsformton s presented n Secton 3 nd soluton of the problem under the mthemtcl Robn boundry condtons pplyng the egenfunctons method s shown n Secton Applcton of the Lplce trnsformton In order to solve the problem ()-(7) we use the Lplce trnsformton wth respect to tme t whch s defned by = st (9) F s F t e dt where s s complex prmeter. The followng property of the Lplce trnsformton of frctonl dervtve wll be used [4] d F t L s F s s F = dt () where <. Applyng the Lplce trnsformton to equton () nd usng the property (), we obtn n ordnry dfferentl equton n the form

4 8 S. Kul, U. Sedlec s [ ] λ d T s T = f x q xs,, x x, x, <, =, () dx The boundry condtons (3), (4) n the trnsform domn re dt λs (, s) = L( TL( s) T (, s) ) () dx dt λs ( x, s) = R( TR( s) T ( x, s) ) (3) dx nd the condtons (5), (6) ssume the form (, ) (, ) T x s = T x s (4) dt dt λs ( x, s) = λs ( x, s) (5) dx dx The generl soluton of the equton () cn be wrtten n the form where P Px Px T ( xs, ) = Ce + Ce + R ( ξ, s) snh( P ( x ξ) ) dξ (6) P x s = nd R ( ξ, s) f ( ξ) q ( ξ, s) x s =. The constnts C, C λ re determned by usng the condtons ()-(5). Assumng: f( x ) = nd q ( xt, ) = for =,, nd solvng the system of four equtons, we obtn the constnts C, C, C, C. After trnsformton, we get Px Px ( P P ) x C = TL( s) ( e ( w)( wr) e ( + w)( + wr) ) + P ( x+ x) TR( s) e w( wl ) M ( P + P ) x Px Px C = TL( s) ( e ( w)( + wr) e ( + w)( wr) ) P ( x+ x) TR( s) e w( + wl ) M

5 Lplce trnsform soluton of the problem of tme-frctonl het conducton n two-lyered slb 9 ( Px ( ) C = T s e + w + L R ( snh cosh )) Px ( ) + ( ) T s e w w Px w w Px M ( R L L ( ) ( w ) P x+ x L R C = T s e (( ) ( ) )) P ( x+ x) T s e + w w coshpx + w + w snhpx M R L L λ, L P where wl = s w L λλ P = λ L s, w P R = λ s, R Px Px ( ( R)( L) ( R)( L) ) Px Px ( ( L+ )( + R) + ( L )( R) ) M = e + w + w + e w w coshpx + w Pλ e w w w e w w w snhpx = s nd Pλ The Lplce trnsform (6) cnnot be nverted nlytclly nd tht s why numercl methods for nverson must be ppled. In the next secton, the soluton for cse of the het conducton problem by pplyng the egenfunctons method s presented nd n the followng secton the numercl results obtned by usng both methods re compred. 4. The method of egenfunctons A soluton to the problem of the het conducton n mult-lyered slb wth frctonl tme-dervtve of the sme order n ech lyer under the mthemtcl Robn boundry condtons s presented by Sedlec nd Kul n the pper [8]. The temperture dstrbuton n the slb by usng the method of egenfunctons hs been obtned. In the cse of the het conducton n two-lyer slb, the formul for temperture n the lyers cn be rewrtten n the followng form: where nd θ T xt, = xt, + Φ x T t + Φ x T t, =, (7) L R θ ( xt, ) = Γ ( t) Ψ ( x), x x x, =, (8), =

6 S. Kul, U. Sedlec Φ Φ x = + x + x = + x x λ λ λ Φ H λ L H R x = + x x = + x x + λ λ λ Φ H λ L H R (9) λ λ λ where H = + x + + x x. L λ R gven by Γ x β t λ E t = fɶ x x dx+ The functons Γ n equton (8) re Ψ, N = x t x ( ) ɶ (, ) + ( t τ) E β ( t τ) q xτ Ψ x dxdτ N,, = x where E, β s the Mttg-Leffler functon [], E = E, nd fɶ x f x x T x T = Φ Φ L R λ d T λ d T qɶ xt q xt Φ x Φ x L R (, ) = (, ) dt dt The functons Ψ, occurrng n equton (8) re gven by () β, λ, = cos, + sn, L Ψ x β x β x β λ Ψ β β, ( x) = A cos ( x x ) B sn ( x x ),,, β, λ () β, λ β, λ β, λ where A = cosβ, x + sn β, x, B = snβ, x cosβ, x L β, λ L nd β, = γ /, wheren γ s -th root of the egenvlue equton β, λ β, λ A B cosβ, ( x x ) A + B snβ, ( x x ) = R R ()

7 Lplce trnsform soluton of the problem of tme-frctonl het conducton n two-lyered slb The coeffcents N occurrng n equton () re gven by ( ) ( cos( )) ( ) sn( ) λ N = β x + u + u β x u β x +,,, β, (( V ) d V ( cos( d) )) ( V ) sn( d) B λ β, (3) where = ( ), β λ A d x x β,, u = nd V =. L B The temperture dstrbuton n the two-lyered slb s completely specfed by equtons (7)-() nd (3) where the egenvlues γ re roots of equton (). 5. Numercl exmples The Lplce trnsform of the temperture dstrbuton n two-lyered slb for dfferent models of the frctonl het conducton s gven by equton (6). The nverse Lplce trnsform wll be obtned numerclly. For numercl nverson of the Lplce trnsforms method wll be ppled whch used the followng formul [] f ( t) f t = lm n (4) n n nlog n n log t n j= j. t j where f ( t) = ( ) f ( n+ j) n The results obtned by numercl nverson of the Lplce trnsform re compred wth the results computed by usng the formul whch s obtned by the method of egenfunctons for the cse of het conducton n the two-lyered slb under the mthemtcl Robn boundry conductons. The numercl clcultons were performed for the slb, whose outer boundres re t: x =, x =.4m nd the nterfce s t x =.m. The het trnsfer coeffcents re ssumed s: L=., R 6 = 6. W/(m C), the therml dffusvtes: 6 = 3.35, = 5.4 m /s, nd the therml conductvtes: λ = 6., λ = 4. W/(m C). The ntl temperture T nd the mbent temperture T R were constnts: T =, T R =. The mbent temperture T L ws functon of tme: T t = A+ B νt where A = C, B = 5 C nd ν = π 5s. Numercl L sn, clcultons were crred out usng the Mthemtc pcge.

8 S. Kul, U. Sedlec Comprson of tempertures n the slb t x = x for vrous vlues of tme ɶt nd order obtned by numerclly nvertng Lplce trnsform nd by egenfunctons method Tble tɶ Temperture [ C] Method of Lplce trnsform Method of egenfunctons T5,x Fg.. Temperture n the slb s functon of xɶ = x x t the non-dmensonl tme t ɶ =5 for dfferent orders =,75;,85;,9;,95;, nd =. x The tempertures of the slb t x= x computed by usng the method of the Lplce trnsform nd the method of egenfunctons for vrous vlues of the order (constnt nd the sme n both lyers) nd for vrous non-dmensonl tme tɶ = t x / re tbulted n Tble. Comprson of the tempertures shows good greement of the results obtned by usng the two methods.

9 Lplce trnsform soluton of the problem of tme-frctonl het conducton n two-lyered slb 3 The temperture dstrbuton n the two-lyered slb under the physcl Robn boundry condtons s functon of xɶ = x x t tme t ɶ = 5 s presented n Fgure. The het conducton n the lyers s chrcterzed by vrous orders of tme-dervtve n the het equton: =,75;,85;,9;,95;, nd =.. The remnng dt re the sme s those presented bove. The curves n Fgure show tht the rto of the frctonl orders n the lyers s sgnfcnt to the het conducton process n the slb. 6. Conclusons The frctonl het conducton n two-lyered slb under the physcl Robn boundry condtons ws consdered. A soluton of the problem by the Lplce trnsformton hs been obtned. The nverse of the Lplce trnsform ws numerclly determned. Good greement shows comprson of the numercl results obtned by numerclly nvertng the Lplce trnsform nd by the method of egenfunctons pplyng to the het conducton problem n slb under the mthemtcl Robn boundry condtons. Becuse the numercl nverson of the Lplce trnsform s n ll condtoned problem, n pplcton of testng method s requred. Although the presented results referred to the frctonl het conducton n the two-lyered slb, the method cn be ppled to the problems of the frctonl het conducton n mult-lyered slbs. References [] Hj-Sheh A., Bec J. V., Temperture soluton n mult-dmensonl mult-lyer bodes, Interntonl Journl of Het nd Mss Trnsfer, 45, [] Özş M. N., Het Conducton, Wley, New Yor 993. [3] Povsteno Y., Lner Frctonl Dffuson-wve Equton for Scentsts nd Engneers, Brhuser, New Yor 5. [4] Povsteno Y., Frctonl het conducton n sem-nfnte composte body, Communctons n Appled nd Industrl Mthemtcs 4, 6,, e-48. [5] Povsteno Y., Fundmentl solutons to tme-frctonl het conducton equtons n two jont hlf-lnes, Centrl Europen Journl of Physcs 3,, [6] Povsteno Y., Frctonl het conducton n n nfnte medum wth sphercl ncluson, Entropy 3, 5, [7] Povsteno Y., Frctonl het conducton n nfnte one-dmensonl composte medum. Journl of Therml Stresses 3, 36, [8] Sedlec U., Kul S., A soluton to the problem of tme-frctonl het conducton n multlyer slb, Journl of Appled Mthemtcs nd Computtonl Mechncs 5, 4(3), 95-. [9] Podlubny I., Frctonl Dfferentl Equtons, Acdemc Press, Sn Dego 999. [] Vlo P.P., Abte J., Numercl nverson of -D Lplce trnsforms ppled to frctonl dffuson equtons, Appled Numercl Mthemtcs 5, 53, [] Wng Q., Zhn H., On dfferent numercl nverse Lplce methods for solute trnsport problems, Advnces n Wter Resources 5, 75, 8-9. [] Povsteno Y., Kleot J., The Drchlet problem for tme-frctonl dvecton-dffuson equton n hlf-spce, Journl of Appled Mthemtcs nd Computtonl Mechncs 5, 4(),