Heat conduction in a composite sphere - the effect of fractional derivative order on temperature distribution

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Willis Ellis
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1 MATEC We of Confeence 57, 88 (8) MMS 7 hp://do.og/.5/mecconf/85788 He conducon n compoe phee - he effec of fconl devve ode on empeue duon Uzul Sedlec,*, Snłw Kul Inue of Mhemc, Czeochow Unvey of Technology, Częochow, Polnd Ac. The m of he conuon n nly of me-fconl he conducon n phee wh n nne he ouce. The ojec of he condeon old phee wh phecl lye. The he conducon n he old phee nd phecl lye govened y fconl he conducon equon wh Cpuo medevve. Mhemcl (clcl) o phycl fomulon of he Ron oundy condon nd he pefec conc of he old phee nd phecl lye umed. The oundy condon nd he he flux connuy condon he nefce e expeed y he Remnn-Louvlle devve. An exc oluon of he polem unde mhemcl condon deemned. A oluon of he polem unde phycl oundy nd connuy condon ung he Lplce nfom mehod h een oned. The nvee of he Lplce nfom y ung he Tlo mehod e numeclly deemned. Numecl eul how he effec of he ode of he Cpuo nd he Remnn-Louvlle devve on he empeue duon n he phee. Keywod: fconl he conducon, heng ouce, Ron oundy condon Inoducon The fundmenl of he clcl he nfe heoy he Foue lw whch led o he polc pl dffeenl equon of he he conducon []. A conequence of he Foue' lw unelc peed of he flow n he medum. Th nconvenence cn e voded y genelzon of he Foue lw whch led o fconl he conducon equon []. The he conducon govened y he fconl dffeenl equon he ujec of ppe [3-9]. Applcon of fconl ode clculu e peened n oo [-] nd ppe [3-5]. If he he nfe n ounded medum condeed hen he he equon complemened y oundy condon. The Dchle, Neumnn nd Ron oundy condon e ofen ued n decng he he nfe eween he ody nd he uoundng. In he clcl he heoy, he Neumnn nd Ron oundy condon nclude he noml devve he oundy of he condeed egon. Inoducng he me-fconl devve n he Neumnn nd Ron oundy condon, he phycl fomulon of hee condon oned [6]. * Coepondng uho: uzul.edlec@m.pcz.pl Revewe: Ján Vvo J., Bnlv Foe The Auho, pulhed y EDP Scence. Th n open cce cle dued unde he em of he Ceve Common Auon Lcene 4. (hp://cevecommon.og/lcene/y/4./).

2 MATEC We of Confeence 57, 88 (8) MMS 7 hp://do.og/.5/mecconf/85788 A oluon of he lne fconl dffeenl equon unde clcl oundy condon cn e deemned n n nlycl fom. To olve he fconl equon unde phycl oundy condon, he Lplce echnque cn e ppled. Th ppoch led o oluon of he polem n he Lplce domn. The empeue duon n he me domn y ung n lgohm fo numecl nveon of he Lplce nfom cn e oned. The mehod of numecl nveon of he Lplce nfom ued n clcl nly cn e lo ppled o Lplce nfom oned y olvng he polem wh fconl devve. Seleced mehod of numecl nveon of he Lplce nfom e peened n ppe [7-]. In h ppe, we peen he oluon of he fconl he conducon polem n phee conng of n nne old phee nd phecl lye. The mhemcl nd phycl fomulon of he Ron oundy condon condeed. The pefec heml conc of he nne phee nd he phecl lye umed. The effec of he fconl ode on he empeue duon n he phee h een numeclly nveged. Fomulon of he polem We conde he me-fconl dl he conducon polem n phee. The wo egon of he phee e dnguhed: - old nne phee nd - phecl lye, whee he dl coodne. The he nfe n he egon govened y he fconl he conducon equon [3]: T T,,, g () g he volumec e of he geneon, he heml dffuvy, he heml conducvy nd denoe he fconl ode of he Cpuo devve wh epec o me. The Cpuo devve defned y [] whee, C d m d m, f m D f f m d, m m f m N () whee denoe he gmm funcon. The oundy condon nd he connuy condon nefce e umed n fom wh he Remnn-Louvlle fconl devve D whch defned y [] RL d f DRL f d, d (3) On he oue ufce of he phee, he Ron oundy condon [6] umed T DRL T T,, (4)

3 MATEC We of Confeence 57, 88 (8) MMS 7 hp://do.og/.5/mecconf/85788 whee he oue he nfe coeffcen nd T he men empeue. The pefec heml conc he nefce eween he nne phee nd he phecl lye deced y condon:,, T T (5) T T DRL DRL,, (6) nd he nl condon F T,. (7) The condon (4) nd (6) fo nd e clled he phycl condon [6], f, hee e o clled mhemcl condon. In he econd ce, he D RL n equon (4) nd (6) men n deny opeo nd cn e omed. We fuhe,. conde he ce of 3 Soluon o he polem In ode o nfom he he conducon equon () no he fconl equon wh U, gven y he followng conn coeffcen, we noduce new funcon elonhp U, T, T,, (8) Tng no ccoun equon (8) n he nl-oundy polem () nd (4-7), we U, n he fom on fomulon of he polem fo he funcon U, * U, g,,, U DRL U U,,, (9) (),, U U () U, U, DRL U, DRL U, () U, F T,, (3) Moeove, he condon (-) e complemened y condon fo, whch oned ung equon (8) 3

4 MATEC We of Confeence 57, 88 (8) MMS 7 hp://do.og/.5/mecconf/85788 * The funcon g, U, (4) n equon (9) e gven y he fomul d T * g, g, (5) d The oluon of he nl-oundy polem (9-4) fo mhemcl nd phycl fomulon wll e peened elow. 3. Mhemcl fomulon of oundy nd connuy condon The he conducon polem (9-4) unde mhemcl condon fo cn e olved nlyclly. We ech fo he oluon o h polem n he fom of he ee of ohogonl funcon :, U,,, (6), In he f ep, we fnd he funcon polem d, y olvng he followng egenvlue,,, d (7), (8) (9),, d d d d d d () () The funcon, e whee, whee B n (),,, A co B n (3),,,,, nd e oo of he equon Q n Q Q (4) 3 4

5 MATEC We of Confeence 57, 88 (8) MMS 7 hp://do.og/.5/mecconf/85788 nd Q co n Q n co Q co n. 3 Thee funcon fulfl he ohogonly condon n he fom fo ' d d N fo ',, ',, ' (5) The coeffcen B,, A,, B,, occung n equon (-3), e deemned y ung condon (9-). Aumng B,, we on A, n, nd B Q., 3, The funcon fe fconl equon whch oned uung equon (6) no equon (9) nd ung ohogonly condon (5). The nl condon fo he funcon oned n ml wy y ung (6) nd (5) n condon (3). The fconl dffeenl equon nd nl condon hve he fom d d * * N,,, g d g, d (6), F T d F T, d (7) N whee N, d, d. A oluon of he polem (6-7) gven y [], *,, g E, dd N * g, E dd,, E,, F T d F T, d N (8) whee z E he Mg-Leffle funcon defned y 5

6 MATEC We of Confeence 57, 88 (8) MMS 7 hp://do.og/.5/mecconf/85788 E z z (9) Fnlly, ng no ccoun equon (8) nd (6), we on he empeue duon n he phee unde mhemcl fomulon of he oundy nd connuy condon n he fom If whee funcon, T, T,, (3) e gven y (8) nd, n, T T con nd g, G con, e gven y F T con funcon e defned y () nd (3). g,, hen he G n,, co, E N, Tn T n,, co E, N,, A,, n, co, n B,, co,,, (3) 3. Phycl fomulon of oundy nd connuy condon A oluon of he he conducon polem (9-4) unde phycl oundy nd connuy condon (, n equon () nd ()) wll e oned y ung he Lplce echnque. The Lplce nfom defned whee f f e d (3) f fo gven funcon of he exponenl ype nd complex pmee. Afe pplyng he Lplce nfomon o he equon (9-) nd (4), nd ung he popee of he Lplce nfom, we on du U, d h (33) U, (34),, U U (35) du, du, U, U, d d (36) 6

7 MATEC We of Confeence 57, 88 (8) MMS 7 hp://do.og/.5/mecconf/85788 du, U, d (37) whee h, T F g,. The oluon of he equon (33) fo, ung he condon (34), cn e wen n he fom U, B nh S h u, nh S udu (38) S nd he genel oluon of equon (33) fo, follow S U, A coh S B nh S h u, nh S udu (39) / whee S. Ung condon (35-37), yem of lne equon wh epec o unnown conn B, A nd B oned Bnh SA P (4) B S S S A B S P coh nh A Scoh S S coh S S nh S B S coh S S nh S S nh S P 3 (4) (4) whee P h u, nh S udu, S S, coh, nh, P S h u S u du h u S u du P3 S hu, coh S udu h u, nh S u du. Suung he deemned conn B, A nd B no equon (38-39), we ge he complee oluon of he polem n he Lplce domn. The empeue duon n he phee gven y 7

8 MATEC We of Confeence 57, 88 (8) MMS 7 hp://do.og/.5/mecconf/85788 T, T L U,,, (43) The nvee of he Lplce nfom of he funcon U,,, clculed numeclly. The fxed Tlo lgohm o numecl nveon of he Lplce nfom h een ued, [8-9]. Applyng h lgohm, he ppoxme vlue of he funcon U, L U, e deemned ung he fomul M p U, U, pexp p Re exp U, j M (44) whee p cg j, cg cg, p M / 5 nd / M, M nume of pecon decml dg. 4 Reul of numecl clculon, j, The effec of he fconl ode of me-devve occung n he he conducon equon nd oundy condon on he empeue duon n he phee h een numeclly nveged. Fo he he conducon model wh he mhemcl fomulon of he oundy nd connuy condon, he eul oned y ung he numecl nveon of he Lplce nfom h een comped wh he exc oluon. The clculon h een pefomed fo he followng geomecl nd heml d: oue du of he phee. m, he du of he nne phee ˆ.6, he heml 6 dffuve e 3.35 m, m, he heml conducve e 6W mk, 54W mk, he oue he nfe coeffcen 5 W m K, he men empeue o T 5 C nd he nl empeue umed Tn o 5 C. Tle. The non-dmenonl empeue ˆ ˆ, ˆ T fo ˆ., clculed y ung he exc oluon nd numecl nveon of he Lplce nfom (NILT) ˆ Exc NILT Exc NILT Exc NILT Exc NILT

9 MATEC We of Confeence 57, 88 (8) MMS 7 hp://do.og/.5/mecconf/85788 The non-dmenonl empeue T ˆ TT n n he eleced pon of he phee fo he mhemcl oundy nd connuy condon (., ) fo dffeen ode of he Cpuo devve e peened n Tle. The volumec e of 3 he geneon n he nne phee g [ W m ] nd n he phecl lye g. The eul peened n he Tle fulfl he condon: Exc NILT Exc The mll dffeence of he Exc nd NILT eul llow fo he ue of he lgohm of numecl nveon of he Lplce nfom o he he conducon polem unde phycl fomulon of oundy nd connuy condon T.5,.4 T.5, c.3 d T.75,... T.,... MC,.7 PC,.7 MC,.85 PC,.85 MC,. PC, Fg.. The non-dmenonl empeue ˆ ˆ, ˆ T funcon of me ˆ fo vou vlue of fconl devve nd : () ˆ.5 ; () ˆ.5 ; (c) ˆ.75 ; (d) ˆ. The me-hoe empeue n eleced pon of he phee e peened n Fg.. The clculon wee pefomed fo dffeen ode of he devve occung n he he equon. In he polem unde mhemcl condon (MC) one w umed.7;.85;. nd. nd fo he polem unde phycl condon (PC) he clculon wee pefomed fo.7;.85;.,.9 3 nd. The volumec e of he geneon w umed : g 5[ W m ] nd g. A w expeced, he dffeence eween he empeue oned fo vou ode of decee f he dnce fom he ouce ncee. The gnfcn effec on 9

10 MATEC We of Confeence 57, 88 (8) MMS 7 hp://do.og/.5/mecconf/85788 he empeue duon n he phee h ode of he devve n he he conducon model. Concluon The polem of he fconl he conducon n phee wh nne he ouce unde mhemcl nd phycl Ron oundy nd connuy condon y ung he Lplce nfom echnque h een olved. I w noed hgh geemen of he eul oned on he of he exc oluon nd he eul compued y ung numecl nveon of he Lplce nfom fo he he conducon unde mhemcl condon. The effec of he ode of he Cpuo devve occung n he he conducon equon on he empeue duon n he phee h een numeclly nveged. I w ed h he empeue n he phee oned fo model wh he me fconl ode wh mhemcl nd phycl Ron condon dffe lghly. The gnfcn dffeence n he empeue hve een oeved fo dffeen ode of he fconl devve n he he conducon equon. Refeence. M.N. Özş, He conducon. (Wley, New Yo, 993). Y. Poveno, Fconl he conducon n em-nfne compoe ody. Communcon n Appled nd Indul Mhemc 6 (), e-48 (4) 3. Y. Poveno, Fconl he conducon n n nfne medum wh phecl ncluon. Enopy 5, (3) 4. Y. Poveno, J. Kleo, The fundmenl oluon o he cenl ymmec mefconl he conducon equon wh he opon. Jounl of Appled Mhemc nd Compuonl Mechnc 6 (), - (7) 5. S. Bl, Tme-fconl he nfe equon n modelng of he non-concng fce el. Inenonl Jounl of He nd M Tnfe, (6) 6. T.H. Nng, X.Y. Jng, Anlycl oluon fo he me-fconl he conducon equon n phecl coodne yem y he mehod of vle epon. Ac Mechnc Snc 7 (6), 994- () 7. S. Kul, U. Sedlec, Lplce nfom oluon of he polem of me-fconl he conducon n wo-lyeed l. Jounl of Appled Mhemc nd Compuonl Mechnc 4 (4), 5-3 (5) 8. S. Kul, U. Sedlec, An nlycl oluon o he polem of me-fconl he conducon n compoe phee. Bullen of he Polh Acdemy of Scence Techncl Scence 65 (), (7) 9. U. Sedlec, Rdl he conducon n mullyeed phee. Jounl of Appled Mhemc nd Compuonl Mechnc 3 (4), 9-6 (4). T. M. Ancovć, S. Plpovć, B. Snovć, D. Zoc, Fconl Clculu wh Applcon n Mechnc. (John Wley & Son, New Yo, 4). A. A. Kl, H. M. Svv, J. J. Tujllo, Theoy nd pplcon of fconl dffeenl equon. (Eleve, Amedm, 6). J.S. Lezczyń, An Inoducon o Fconl Mechnc. The Pulhng Offce of Czeochow Unvey of Technology, Czeochow ()

11 MATEC We of Confeence 57, 88 (8) MMS 7 hp://do.og/.5/mecconf/ M. Dl, M. Bhou, Applcon of fconl clculu. Appled Mhemcl Scence 4 (), -3 () 4. A. Dzelń, D. Seocu, G. Sw, Some pplcon of fconl ode clculu. Bullen of he Polh Acdemy of Scence Techncl Scence 58 (4), () 5. W. E. Rln, Applcon of fconl ode heoy of hemoelcy o D polem fo phecl hell. Jounl of Theoecl nd Appled Mechnc 54 (), (6) 6. Y. Poveno, Tme-fconl he conducon n n nfne medum wh phecl hole unde Ron oundy condon. Fconl Clculu & Appled Anly 6 (), (3) 7. K. L. Kuhlmn, Revew of nvee Lplce nfom lgohm fo Lplce-pce numecl ppoche. Numecl Algohm 63 (), (3) 8. J. Ae, P. P. Vló, Mul-pecon Lplce nfom nveon. Inenonl Jounl fo Numecl Mehod n Engneeng 6, (4) 9. B. Dngfelde, J. A. C. Wedemn, An mpoved Tlo mehod fo numecl Lplce nfom nveon. Numecl Algohm 68, (5). H. Sheng, Y. L, Y. Chen, Applcon of numecl nvee Lplce nfom lgohm n fconl clculu. Jounl of he Fnln Inue 384, (). I. Podluny, Fconl dffeenl equon. (Acdemc Pe, Sn Dego, 999). K. Dehelm, The nly of fconl dffeenl equon. (Spnge-Velg Beln Hedeleg, )

Calculus 241, section 12.2 Limits/Continuity & 12.3 Derivatives/Integrals notes by Tim Pilachowski r r r =, with a domain of real ( )

Calculus 241, section 12.2 Limits/Continuity & 12.3 Derivatives/Integrals notes by Tim Pilachowski r r r =, with a domain of real ( ) Clculu 4, econ Lm/Connuy & Devve/Inel noe y Tm Plchow, wh domn o el Wh we hve o : veco-vlued uncon, ( ) ( ) ( ) j ( ) nume nd ne o veco The uncon, nd A w done wh eul uncon ( x) nd connuy e he componen