The Solution of Heat Conduction Equation with Mixed Boundary Conditions

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Preston Wilson
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1 Joul of Mhec d Sc (: , 6 ISSN Scece Publco The Soluo of He Coduco Equo wh Mxed Bou Codo Ne Abdelzq Dee of Bc d Aled Scece, Tfl Techcl Uvey PO Box 79, Tfl, Jod Abc: The u devoed o deee oluo fo o-oy he equo xl yec cyldcl coode ude xed dcouou bou of he f d ecod kd codo, wh he d of Llce fo d eo of vble ehod ued o olve he codeed oble whch he dul egl equo ehod Key wod: Nooy he equo, dul egl equo, xed bou codo INTODUCTION The ehod of dul egl equo wdely ued o olve ellc l dffeel equo wh y hycl d echcl lco [-4], lo evel echque wee develoed he l ffy ye o olve dul equo wh dffee coode ye dcu I h u he oluo of wodeol o-oy he coduco oble xlly yecl cyldcl coode wh dcouou xed bou codo f d ecod kd o he level ufce of e-fe old cyldcl coode wll be dcued The oluo of he oble bed o he lco of dul egl equo ehod wh he hel of he Llce fo d eo of vble I kow h he oluo of dul egl equo oduced o oe ye of gul egl equo of he f kd wh ukow fuco, wegh d fee e deed o he ee of Llce fo The exc oluo of uch egl equo c be obed by exeg ukow fuco he fo of fucol ee owe of Llce fo ee The gol of gve oble h wok o exed he ue of dul egl equo ehod o olve bolc l dffeel equo wh xed dcouou bou codo, by ug oe dcouou egl echque Th echque led o olve dffee ye of dul equo eled o dffco heoy, elcy heoy d ohe lco [3,4], whe he ecod oe of dul egl equo hoogeeou I he coduco heoy, hee e evel ehod wee ued fo olvg he coduco oble ude uxed bou codo oed ou [5-7] Mdk educed oe dul equo o he Fedhol egl equo of he ecod kd [8,9] MATHEMATICAL FOMULATION OF THE POBLEM The of h u o olve he ooy he coducvy dffeel equo fo hlf-ce cyldcl coode wh xlly yey T T T T + + ( z τ whee T T (, z, τ he eeue dbuo fuco, < <, < z < e he coeodg cyldcl coode, τ > e, he eeue coducvy coeffce (co The equo ( c be olved by ug he codo T T T z ( ude xed dcouou bou codo f d ecod kd o he level ufce z T(,, τ f (, τ, S, (3 T (,, τ / z f (, τ, S, (4 whee S (, S (, The l codo T (, z, (5 The ukow fuco f, f (3,(4 couou d hve he led vo wh eec of ech vble d τ, oeove f (, τ d < f (, τ dτ <,, Thee eco llow o ly Llce fo wh eec o τ d Hkle fo wh eec o oeove, we ue h he fuco f(, τ,, hve boluely couou devve wh eec o The hycl gfcce of he gve bou vlue oble h, de he dk < <, o z, he eeue fuco gve by T (,, τ f(, τ d oude he dk < < z he flow gve by Tz (,, τ f(, τ Nex, o lfy he vego of he oluo, we wll ue del ulo ex, e, f Coeodg Auho: D Ne Abdelzq, Dee of Bc d Aled Scece, Tfl Techcl Uvey, PO Box 79, Tel: , Tfl Jod, Fx: 533, 346

2 J Mh & S, (: , 6 SOLUTION OF THE POBLEM The bou-vlue oble eoed bove hould be olved by lyg of he Llce fo he τ -vble Defg of T(, z, [] T (, z, T (, z, τ ex( τ dτ Tkg he Llce fo d eo of vble o equo ( wh egd o codo (, he geel oluo of he oble obed fo of oe egl (,, (, ex( + α T z u J d (3 whee J he Beel fuco of he f kd of ode zeo, he ee of eo of vble, he ee of Llce fo wh α / e( > Alyg he Llce fo o he bou codo (3,(4 d ug hee codo o he geel oluo (3, we ob he followg dul egl equo o deee he ukow fuco u(, u(, J( d f(,, S (3 u(, + α J ( d f(,, S (33 I geel equo (3, (33 c be olved by ug dcouou egl echque, f he ecod equo hoogeeou, e, f (,, hu o lfy he vego of he oluo, we wll ue del ulo ex, e f I geel whe he ecod dul egl equo o-hoogeou, c be educed o he hoogeeou by exo f (, Hkle egl fo (, (, + α f F J d, (34 Whee F(, kow fuco deee by he veo Hkle fo / F(, y f ( y, J ( y + Nex we dow u(, F(, A(,, he ue he exo (34, he followg dul egl equo e obed o deee he ukow fuco A(, A(, J ( d D(,, S A + J d S (, /, whee D(, f (, F(, J ( d A, he dul egl equo (3 d (33 ed o he dul egl equo of he fo u( J ( d f, S, u( J ( d f, S Moe del, dcued oogh [,3] To lfy he oluo of equo (3,(33,, we wll code ecl ce uch h f (,, f (, f /,hu, he dul equo (3,,(3,3 wll ke he fo u(, J( d f /, S (35 u(, + α J( d, S (36 elcg he fuco u(, by ohe ukow fuco (, wh he hel of he elo (, (, co( + α + α u d (37 I ued (, dffeeble wh eec o, whee (,, lo he vee Llce fo L [ (, ] (, τ ex, fuhe,we ue h (, τ couou o ecewe couou y evl τ < τ < τ fo τ > d τ (, τ bouded + τ,oeove (, τ exoel ode ely ex( γτ (, τ bouded fo oe ove ube γ τ [5] Subug (37 o (36 d egg wh eec o fo o, he echgg he ode of ego, ug he vlue of he dcouou J( ( + α egl [,] + α x d, > x, co ( x α, x >, x eue he equly (36 o zeo, uch h ( ( (, (, / ( + α J( (, d + / J ( ( + α (, d d + α 347

3 J Mh & S, (: , 6 Subug (37 o (35 d ug he dcouou egl [] ( + α J( co d + α ( α < < <, ex ( ( α < < < A f kd gul egl equo obed o deee he ukow fuco (, ( (, ex ( / d (, ( / d f ( /, S (38 Sce (, lycl fuco of ee, / c eee fucol ee owe of degee (, ex( α / (39 The vee Llce fo fo (39 ex [8] H ( L L (, (, τ ex( L, / τ L 4τ H L Hee fuco [3] Mullyg equo (38 by ex( α d exdg ex( x d ( x (38 oe Mclu ee, he ug exeo (39,he followg egl equo obed o deee he ukow ebe,,, ( +!( + + ( ( +!( + l l / f ( d ( d (3 l l! Equg he coeffce of he lef d gh hd / de (3 equl owe of we fd h A : l ; {(,(,}, we ge Abel egl equo o evlue d f (3 Solvg Abel egl equo (3 yeld d yf ( y d y / A : l ;{(,(,,(,,(,}, Abel egl equo obed o deee d d f! (3! Ug he veo fo deeg (3, o h he oluo gve by d y f ( y ( d d + y!! A : l, {(, (,,(,,(,,(,}, l oce (3 d (3 he oluo fo gve by d y d y y ( f ( y ( y d + ( d (!!(! Fo he bove evluo of,,, ey o coclude h ( fy he followg ecue foul, fo eve dex ( [ / ] ( k d + ( ( k k (k ( k ( d [ / ] / k k (k ( ( k! ( ( (k! (33 k (k ( d f ( y! The oluo Abel egl equo (33 fo he ukow fuco ( e of he kow fuco k, ( k gve by: d y d y f ( y +! ( [ / ] / k k ( (k ( ( k k ( (k! [ / ] ( k y ( (k ( ( k ( ( y d k k ( ( k! (34 y d O he ohe hd fo odd vlue of we hve he ecue foul ( d (k ( + (k k ( (k! ( ( (k + k (k +! k k (k ( d ( k (k + d f ( y, (35 (! 348

4 J Mh & S, (: , 6 Ag eg (35 Abel egl equo o deee ( fo odd dex d y d y ( f ( y +! k ( k ( (k ( ( k + + k (k +! (k y ( k ( (k ( ( y d k k ( (k! y d (36 I cul,f he eeue fuco de he ego <<, z co, y f (, K / equo (3 he he vlue of ech,,, olyol K ( K ( + + ( + K! (!(!( I geel f he fuco f (, deed o e( geel ce, hould be exeed / j, j f (, f j hece, he gh hd de of (3 c be we l ( l + j / f j l j l! Equg he lef d he gh hd de of (3 o deee ecuece elo o ( fo eve d / odd dex owe of l j + To deee he geel oluo T(, z,, ubug equo (34 d (36 o equo (39, he ug he obg eul o (37 d flly ubug he eul o he geel oluo (3 we ge / T (, z, ex α ( + ( + J( co α ex z α d d + α (37 Now exe (37 co( x (ex( x + ex( x /, d ue he vlue of he egl [] ex( x + y ex( y + c x J ( cx dx (37- x + y + c The oluo of (37 wh egd o (37- c be we T (, z, ex( α / ex( α ex( α + d (38 whee ( z +, ( z + + I cle h d z ed o fy T(, z, vhed The vee Llce fo of (38 T (, z, τ τ / { ex( } + ex( whee L H L L H L d ( + L, L 4τ ( + 4τ Ioduco he oluo of dul egl equo (35 d (36 o egl equo of he ecod kd: We wll dcu ohe echque fo olvg he bove dul equo (35 d (36 by educg hee equo o ecod kd gul egl equo ewg equo (38 he followg fo ug he elo ch( x h( x ex( x (, ch ( / d ( / (, d f (,, S (4 The ex e exdg (4 ch( x, h( x, oe Mclu ee, he egg he obg eul, le clculo yeld f kd egl equo o deee he ukow fuco (, (, + / d (, ( d (! ( + / (, ( d f (,, S + / ( +! (4 Teg (4 Able egl equo by lyg he veo fo (,, he echgg he ode of ego de he egl g fo he obg eul, ecod kd gul egl equo cheved (, F(, (, N(,, d + (, M (,, d (43 d yf ( y whee F(, d y + / ( d y( y + (! M (,,, + d y / 349

5 J Mh & S, (: , 6 / d y( y N (,,! d y Subug he exo (39, o equo (43 egl equo of he ecod kd obed o evlue he equece,,,,, + l l / / F l / l! + + ( ( +! + / + M (, d ( N (, d! (44 whee d y( y M (, d y d F (,,3/, / d F (,,3/, / 4 F (,,5/, / 3 Whee F hyegeoec fuco [4] d / d y( y N(, d d y d Γ (/ + Γ (/ + ( (, d Γ ( + Γ ( + Γ ( x g fuco [4] If we eque he lef d he gh hd de of equo / (44 wh equl owe of, he e ecue elo o fd,,,,, gve (34 d (36 The e,,,,, u be fy he oey C (, C (, CONCLUSION Wh he hel of kow ehod, he oluo of o-oy he coduco equo ude xed bou codo obed by oducg he gve oble o oe ye of dul egl equo wee olved by ug dcouou egl echque d he kow geeg ee (, ex( α / If he Llce fo ee e dul equo ed o zeo, he oluo of he codeed oble oduced o he kow eul The bove exc oluo of he xed bou vlue oble gve fo of fe ee c be ued wdely o olve vou xed bou oble del wh ue e he equo fo exle o fe o fe cylde, uyecl cyldcl coode, hecl coode d ohe xed oble EFEENCES Mdl, BN d N Mdl, 999 Advce Dul Iegl Equo Lodo, CC Chkb, A d N Mdl, 998 Soluo of oe dul egl equo ZAMM Z Agew Mh Mech, 78: Sedo, I, 966 Mxed bou vlue oble oel heoy Noh Hol Pub Aeed 4 Fbc,VI, 99 Mxed bou vlue oble oel heoy d he lco egeeg Kluwe Acdec Publhe 5 Ozk MN, 98 He Coduco Wly d So, Ic 6 Hol, JP, He Tfe McGw-Hll Co NewYok 7 Kozlov, VP d PA Mdk, Sye of egl d dffeel equo L-ee oble of hecl hyc d defco ehod of he chcec BGU Pub Mk-Belu 8 Mdk, PA, The ehod of he dul egl equo fo ly of he fe ocee Mhecl Modelg d Aly, 6: Mdk, PA, The oluo of he equo wh xed bou codo o ufce of ooc hlf-ce Dffeelye Uvey, 37: 38-4, ( u Co, M (ed, 988 The egl fo ehod hel d flud cece d egeeg New Yok, Begel Houe, Ic Gdey, IS d IM yzk, 99 Tble of egl, ee d oduc Acdec Pe, New Yok Podcov, AP, U Bchcov d OI Mchev, 983 Iegl d See of Secl Fuco Mocow, Nwk 3 Tee, NM, 996 Secl fuco oduco o he clcl fuco of hecl hyc Jho Wley, New Yok 4 vlle, ED, 97 Secl Fuco Chele P Co, New Yok 35

Laplace Transform. Definition of Laplace Transform: f(t) that satisfies The Laplace transform of f(t) is defined as.

Laplace Transform. Definition of Laplace Transform: f(t) that satisfies The Laplace transform of f(t) is defined as. Lplce Trfor The Lplce Trfor oe of he hecl ool for olvg ordry ler dfferel equo. - The hoogeeou equo d he prculr Iegrl re olved oe opero. - The Lplce rfor cover he ODE o lgerc eq. σ j ple do. I he pole o