GENERALIZATION OF THE FINITE INTEGRAL TRANSFORM METHOD

Transcription

1 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, IX.5 GENERAIZATION OF THE FINITE INTEGRA TRANSFORM METHOD NEW VERSION Fall 8: IX.5. DERIVATION OF THE FINITE INTEGRA TRANSFORM OD VERSION Fall 7: IX.5. Itodctio 864 IX.5. Slemetal Stm-ioville Poblem 864 IX.5.3 Fiite Itegal Tasfom 865 IX.5.4. Oeatioal Poety of the Fiite Itegal Tasfom 865 IX.5.5. Tasiet Heat Tasfe i the Fi 868

2 856 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9 Setembe, 8 BYU, Decembe 8

3 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, IX.5. DERIVATION OF THE FINITE INTEGRA TRANSFORM METHOD NEW VERSION Itodctio et (,ξ ) be a fctio of ad some othe vaiables deoted by a cosolidated symbolξ (which is teated as a aamete). Coside a atial diffeetial eqatio fo (,ξ ) i the domai fo which the vaiable is defied i the fiite iteval [, ]. et be the diffeetial oeato with esect to the vaiable, ad let ξ be the diffeetial oeato with esect to the othe vaiables ξ. The the PDE eqatio fomally ca be witte i the followig oeato s fom: ( ξ ) G (, ) F, ξ, G ξ ξ () Also coside the mied boday coditios give at the boday of G : h f ( ξ ) h f ( ξ ) () (3) itegal tasfom O objective is to costct a itegal tasfom :(, ξ ) ( ξ ) I :(, ξ ) ( ξ ) its alicatio to the PDE elimiates the diffeetial oeato : ( ξ) ( ξ) ( ξ) ( ξ) ( ξ) Q,f,f F ξ I sch that whee Q is some eessio which does ot iclde deivatives with esect to, ad F is a tasfom of the fctio F : I { F} F I the Catesia coodiate system fo diffeetial oeato, sch a itegal tasfom was defied by the itegal eel as the omalized eigefctios of the coesodig slemetal Stm-ioville Poblem (see table SP). Hee, we coside a abitay diffeetial oeato, bt o aoach is also based o the soltio of the coesodig eigevale oblem. Si ξ Fo simlicity, the vaiables ξ will be omitted fom the otatios of (,ξ ). Deivatio will be cocetated oly o the vaiable. Theefoe, otatio ( ) does ot elimiate the esece of the othe vaiables o aametes.

4 858 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9. Diffeetial oeato Coside the secod ode diffeetial eessio defied as a eslt of alicatio C, of the oeato to the fctio, a a a whee coefficiets a C[, ] a > < < (4) ae cotios fctios, ad some eqiemet of diffeetiability will be eeded too. Assme that the followig boday coditios ae imosed o ( ) : α β f ξ, α β f ξ, α > (5a) β α > (5b) β Defie fctios ( ),q( ), ad ( ) by the eqatios a e a d a q a Note, that a >. The oeato ca be ewitte i self-adjoit fom (eecise): Self-adjoit fom a a a ( ) q (6)

5 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, Slemetal eigevale oblem Eigevale oblem y y Coside eigevale oblem fo the give oeato with a aamete sbject to the boday coditios, which ae the homogeeos case of coditios (5): < < (7) β β α y y, α y y, α > (8) α β β > Rewite it sig the self-adjoit fom of oeato ( y ) qy y ad eaage this eqatio to the Stm-ioville fom (see Eq.(5),.438): y q y (9) ( ) Accodig to Stm-ioville Theoem (VI.3,.439), fo this eigevale oblem to be solvable, aamete has to be o-ositive, µ. The thee ae ifiitely may distict vales of aamete µ : µ < µ < µ <... ( µ is a eigevale oly if α α ) fo which eqatio (9) has the coesodig o-tivial soltios y ( ) y µ y () satisfyig the boday coditios (8). Ie odct Defie ie odct fo v( ), (, ) with the weight fctio as v,w v w d () The om of fctios v (, ) itodced om as ca be defied with the hel of Nom v v,v v d () The the vecto sace, with the defied ie odct is the Hilbet sace (Riesz-Fische Theoem, VI.,.43).

6 86 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9 Othogoality of { y ( )} Accodig to Stm-ioville Theoem, the set of eigefctios { } comlete set of mtally othogoal fctios i (, ) y is a ( y,y m) if m (3) ad ay fctio v (,) eigefctios { y ( )} as ca be eeseted by the seies easio ove v cy (4) whee the coefficiets of easios ae c ( v,y) ( y,y ) ( v,y ) y (5) Theefoe, easio of v( ) ca be ewitte as ( v,y ) y v (6) y This eigefctio easio fomla ca be sed fo defiitio of the Fiite Itegal Tasfom, ad fo the coesodig Ivese Tasfom fo ecostctio of the fctio fom its tasfom: 3. Fiite Itegal Tasfom Diect Tasfom { } I y,y d (7) Ivese Tasfom I { } ( ) y (8) y The defied fiite itegal tasfom ai is based o the soltio of the Stm- ioville oblem. Fo alicatio of this itegal tasfom to soltio of the boday vale oblems, the oeatioal oety has to be develoed. The followig elatio will be sed i deivatio of oeatioal oety. Usefl elatio Fom ($) ( µ ) y q y y qy µ y (9)

7 I { } Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, Oeatioal Poety fom Eq.(9) I et ( ) satisfies o-homogeeos boday coditios (5), ad let be the diffeetial oeato defied by eqatios (4), which has the self-adjoit eesetatio (6). Deive the eslt of alicatio of the Fiite Itegal Tasfom to the diffeetial oeato : ( ) q q y d ( ) q y d ( ) y d qy y d ( ) qy y y d qy y y d qy qy y y y d ( y ) qy µ y µ y y qy y d d d d d d qy d y y µ y d qyd qy y y µ d Robi-Robi coditios The case of Robi-Robi boday coditios. Eqatios (8) ca be ewitte as β y y, α >, β > α β y y, α >, β > α I y ( ) ( ) ( ) y ( ) ( ) ( ) y ( ) ( ) ( ) y ( ) ( ) ( ) µ { } β β µ α α ( β β ) µ α α f f y y y y y y y ( ) ( ) α ( ) β ( ) y ( ) ( ) α ( ) β ( ) α µ f f y y ( ) µ α α α f f () Robi-Robi I { } µ y ( ) ( ) y ( ) ( ) α α

8 86 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9 5. OUTINE of the Fiite Itegal Tasfom method ) Give oeato a a a, Boday coditios ae imosed o ( ) : α β f( ξ), α ( ) β ( ) f ( ξ), α α a > < <, β > β > a Defie fctios e a q a a a d ad ewite ( ) q ( ) self-adjoit fom ) Slemetal eigevale oblem (Stm-ioville oblem) y y, < < α y β y, α β > β α y y, α β > Soltios fo µ. Solve the b.v.. fo eigevales µ : µ < µ < µ <... ( µ oly if α α ) ad fo the coesodig o-tivial soltios y ( ) (eigefctios) y µ y satisfyig the boday coditios. Defie ie odct fo v( ), (, ) with the weight fctio v,w v w d ad the om 3) Itegal Tasfom { } v v,v v d I Ivese Tasfom I { } ( ) y,y y y d f f (R-R) 4) Oeatioal oety I { } µ y ( ) ( ) y ( ) ( ) 5) Aly tasfom I { } to the diffeetial eqatio. 6) Solve fo tasfomed fctio α α ad aly the ivese tasfom to fid the soltio I { } as.

9 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, EXAMPE Viscos flid flow betwee two otatig cylides (see IX.4.7.5), SF-AD---FiT eamle.mws f ω ν, < <, t > ν α iitial coditio: (,) ( ) boday coditios:, ( t) c f Nema f t ω Diichlet ( ) y y µ y ) Diffeetial oeato ( ) ) Stm-ioville oblem self-adjoit fom, weight ( ) y ( ) (N) homogeeos y( ) (D) boday coditios 3) Itegal Tasfom 4) Oeatioal oety { } ( µ ) y y y Bessel eqatio of ode ν. Eigevales µ ae the ositive oots of chaacteistic eqatio (.58). y ae (.59): The coesodig soltios y ( ) J ( µ ) J ( µ ) Y ( µ ) Y ( µ ) y( ) y ( ) d { } (,y ) y µ - fy fy I sqaed om (see Ozisi,.4) I 5) Aly tasfom I{ } to eqatio αµ -α f y ( ) α f y ( ) Aly alace tasfom su αµ U α f y ( ) f y ( ) 6) Fid the tasfomed soltio d (N-D,.83 ) s U fy fy µ s s αµ t fy fy e αµ µ U Aly ivese alace tasfom Soltio (by ivese FIT) (,t ) y ( ) y αµ ( e ) f y f y y µ y t Steady state soltio s ( ) Not-oscillatig soltio (,t ) ( ) ( ) f f f f s f y f y y e αµ t s µ y

10 864 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9 7. EXAMPE -D Foe-Plac Eqatio FP-.mws [see class website] a a, < <, t > a >, a (,t) t > (,t) f ( t) (,t) t > (,t) f ( t) δ, S 7. EXAMPE -D Foe-Plac Eqatio Ostei-Uhlebec ocess Rise,., Eq. (5-3) a a ( ), < <, t > a >, a (,t) t > (,t) f ( t) (,t) t > (,t) f ( t) δ, S Shodige [Rise,.7] Semiclassical ase Eqatio [Rise,. 377, Eq.(.7)] E E E P t κ c z ε t

11 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, Itegal tasfoms based o sigla Sigla Stm-ioville Poblems 8. EXAMPE Cicla domai

12 866 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9 9. EXAMPE Hagig Cable (Chai) FP-.-CHAINmws w, < <, t > (,t ) < t > (,t) t >, (,) With damig w γ (,t ) < t >, < <, t > (,t) t > (,t) f ( t),, Oscillatio dive by eteal foce at the boday w, < <, t > (,t ) < t >,t f t h siωt t > (,) (,)

13 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, EXAMPE -D Heat Eqatio i Catesia coodiates to show how the taditioal Fiite Itegal Tasfom is develoed by the geealized aoach

14 868 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9 IX.5. INTRODUCTION et (,ξ ) be a fctio of ad some othe vaiables deoted by a cosolidated symbolξ (which will be teated as a aamete). Coside a atial diffeetial eqatio fo (,ξ ) i the domai fo which the vaiable is defied i the fiite iteval [, ]. et be the diffeetial OD NOTES 7 vesio oeato with esect to the vaiable, ad let ξ be the diffeetial oeato with esect to the othe vaiables ξ. The the PDE eqatio ca be witte i the followig oeato s fom: ( ξ ) G (, ) F, ξ, G ξ ξ () Also let the mied boday coditios be give at the boday of G : h f ( ξ ) h f ( ξ ) () (3) O goal is to costct a itegal tasfom :(, ξ ) ( ξ ) I sch that its alicatio to the PDE elimiates the diffeetial oeato : ( ξ) ( ξ) ( ξ) ( ξ) ( ξ) G,f,f F ξ whee G is some eessio which does ot iclde deivatives with esect to, ad F is a tasfom of the fctio F. Fo the fiite iteval i the Catesia coodiate system with, sch a itegal tasfom was defied by the itegal eel as the omalized eigefctios of the coesodig slemetal Stm-ioville Poblem (see table SP). Hee, we coside a abitay diffeetial oeato, bt o aoach is also based o the soltio of the coesodig eigevale oblem. IX.5.. SUPPEMENTA EIGENVAUE PROBEM Coside a boday vale oblem (-3) fo the PDE which icldes the diffeetial oeato edced to self-adjoit fom: q (4) with fctios, > i (, ). Coside a slemetal eigevale oblem (Stm-ioville Poblem): (, ) y y (5) ( ) h y( ) ( ) h y( ) y (6) y (7) Boday coditios (6-7) ae the homogeeos case of boday coditios (-3).

15 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, Accodig to the Stm-ioville Theoem, thee eist ifiitely may vales of the aamete (eigevales) fo which SP (5-6) has the o-tivial soltio y (eigefctios). Ths, fo the slemetal eigevale oblem we have y y,,... (8) > (i a case of q ad h (Nema b.c.s), h is also a eigevale). IX.5.3 Fiite Itegal Tasfom Defie a fiite itegal tasfom : (, ξ ) ( ξ ) (with the weight fctio aametes ξ ) with a eel d I as a weighted ie odct ) of fctio (si fo simlicity othe (9) whee the eel of the itegal tasfom is defied with the hel of omalized eigefctios of the slemetal eigevale oblem (5-8): y o y () y The the ivese tasfom I : ( ξ ) (,ξ ) y o y is defied by the ifiite seies: () This defiitio of the ivese tasfom ca be teated as the geealized Foie y othogoal with esect to seies based o the comlete set of fctios { } the weight fctio (see Stm-ioville Theoem). IX.5.4. Oeatioal Poety of the Fiite Itegal Tasfom Coside the alicatio of the itegal tasfom I : (, ξ ) ( ξ ) to q (4) Assme also that the fctio (omit fo simlicity the aamete ξ ) is sbject to the boday coditios: ( ) h( ) f (5) ( ) h( ) f (6) whee fctios f ad f ca deed, i geeal, o the aametes ξ. Fist establish the followig eslts: ) Becase the eel of the itegal tasfom is jst a mltile of the eigefctio y, it is also a eigefctio coesodig to the eigevale, ad, theefoe, it satisfies the eqatio o ( ) q Rewite this eqatio as ( ) q (7) also satisfies homogeeos boday coditios of SP ) The eel

16 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9 87 h h Fom which, ovided that coefficiets,, yield h (8) h (9) Aly ow the itegal tasfom I to (4): { } I { } d defiitio (9) d q eqatio (4) d q simlify d d q ead [ ] d d q itegatio by ats [ ] [ ] [ ] d d q itegatio by ats [ ] [ ] [ ]d q d q se eqatio (7) [ ] [ ] [ ]d q d d q ead [ ] [ ] d two last tems cacelled [ ] [ ] defiitio (9) h h [ ] [ ] h h f f Theefoe, the followig oeato s oety is deived { } I f f () If boday coditios fo the PDE (5-6) ae homogeeos ( f f ), the oeato s oety edces to { } I () Theefoe, alicatio of the fiite itegal tasfom I to the PDE elimiates the coesodig diffeetial oeato fom the eqatio.

17 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9 87 Geeal idea of the fiite itegal tasfom method: Give eqatio F (, ξ ) [ ] f (,) ξ, ξ G ξ Rewite i the fom: ( ) q F (, ξ ) ξ [ ] f (,), ξ G ξ Eigevale Poblem: X µ X SP: ( ) X qx µ X X [ µ ] X q X [ X] µ X X X Ie odct (,v) v d Nom X ( X,X ) Oeato is Hemitia (,v) (, v) if [ ] [ ], v Geealized Foie seies cx (,X ) X X X X, (,) X X eel X X Itegal Tasfom T { } (, ) Ivese Tasfom T { } T,,,, fo [ ] Oeatioal Poety { } Thee will be additioal tems if boday coditios ae o-homogeeos: [ ] f

18 87 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9 IX.5.5. Tasiet Heat Tasfe i the Fi Coside heat tasfe fom the eteded sface (fi) oe ed of which is attached to the base at temeate T b ad the whole sface with the othe ed is eosed to the covective eviomet with the coefficiet of heat tasfe h ad temeate T. We assme that: a) Geometically, the system is cosideed to be oe-dimesioal i.e. i ay coss sectio cost temeate is ifom ad is eeseted fo each T,t. momet of time t by b) alog the fi heat is tasfeed by codctio oly (the Foie aw): T Qcod q Ac Ac c) fom the sface of the fi by covectio oly (adiative heat tasfe is egligible o the coefficiet h is a effective coefficiet which icldes cotibtio both fom covectio ad adiatio): qcov ht [ T ] d) thee is o heat geeatio iside a fi; e) the ate of heat stoage i the fi is give by T Qst Vρc ) Deivatio of the goveig eqatio Coside a fiite cotol volme small eogh to be cosideed aoimately as a cylide betwee coss sectios ad fo which we have: A aea of coss-sectio c P( ) eimete As P lateal sface aea eosed to covective eviomet V A volme c Eegy balace fo the cotol volme: Qcod,i Qcod,ot Qcov Qst ( ) Q Q h T,t T A A ρc cod cod s c T Divide the whole eqatio by Qcod ( ) Qcod As T h T (,t) T Ac ρc ad let, at the limit we get As T Qcod h T (,t) T Ac ρc Relace ate of codctio heat tasfe by the Foie aw T,t s Ac h T(,t) T A Ac ρc T The deived eqatio descibes the o-statioay temeate distibtio alog the fi. Also aly the coditios: T, iitial coditio T

19 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, Tb T (,t) T,t Diichlet boday coditio h T,t T Robi (covective) boday coditio et s mae fthe assmtios. Assme the fi to be of costat cosssectio ad the eimete with costat themal codctivity: P P A Ac As As P the P The the goveig eqatio ca be edced with T (,t) T Ac hp T (,t) T Acρ c T (,t) hp ρc T T (,t) T A c cost Itodce the followig otatios: m hp A c ρc a θ,t T,t T ecess temeate θ (,t) T (,t) θ (,t) T (,t) θ,t θ T T b b The we have the followig IBVP fo the ecess temeate: θ m θ a θ θ (,) iitial coditio θ (,t) θb Diichlet θ hθ Robi This eqatio is ot a classical Heat Eqatio becase it icldes a additioal tem with a ow fctio. et s develo the Fiite Foie Itegal Tasfom (9) secifically fo this oblem. ) Rewite the eqatio with the diffeetial oeato i self-adjoit fom θ ( θ ) m θ a θ θ a,

20 874 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9 3) Slemetal Stm-ioville oblem fo oeato : y y y y ( ) hy ( ) Fid eigevales ad eigefctios: solve ODE y m y y y ( ) hy ( ) y ( m ) y Ailiay eqatio ( ) M ± m M m Geeal soltio: ccosh m csih m m > y c c m ccos m c si m m < Aly fist boday coditio: c c m > c m > y c c m c m c c m < c m < The the geeal soltio edces to c sih m m > y c m c si m m < Deivative of geeal soltio is c m cosh m m > y c m c m cos m m < Aly secod boday coditio c m cosh m h sih m m > c ( h) m c ( m cos m h si m ) m < Becase we eed a o-tivial soltio the secod costat caot be eqal to zeo, c. Theefoe, it yields that > < m cosh m h sih m m h m m cos m h si m m The secod eqatio caot be satisfied, becase,h, >. Theefoe, the eigevales have to be the ositive oots of eqatios:

21 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, ae ositive oots of > m cosh m h sih m m m cos m h si m m < eigefctios y itegal tasfom :θ θ The coesodig eigefctios ae sih m m > si m m < 4) Costct the itegal tasfom ai: I θ θ t,t y d y I :θ θ θ(,t) θ ( t) N Oeatioal oety: I { θ } θ θ y N y y d m m > θ θb m m < 5) Tasfomed eqatio I θ a θ m m > θ θb a m m < θ tasfomed eqatio θ θ b θ a a m m > m m < with tasfomed iitial coditio: θ ( ) 6) Soltio of the tasfomed eqatio (vaiatio of aamete): θ ( t) t t t m m > θ a a b e e dt a m m <

22 876 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9 m m > t t t θ b a a e e dt a m m < m m > t t θ b a a e e m m < m m > θ t b a e m m < 7) Soltio of IBVP ivese tasfom: θ θ t b y a,t e N m m < Soltio: m m > Ret to the oigial fctios: m m > t y a T (,t) T ( Tb T) e N m m < The soltio ca be seaated ito steady state ad tasiet ats: m m > y T,t T ( Tb T) N m m < t a e y b N m m < ( T T ) s t m m > Fo this oblem, the steady state soltio is ow to be [Icoea]: h cosh m( ) sih m( ) s T ( Tb T ) m h cosh m sih m m Theefoe, the soltio ca be witte also as h cosh m( ) sih m( ) T (,t ) T ( Tb T ) m h cosh m sih m m t m a m > e y ( Tb T ) N m m <

23 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, ) Male Eamle (GFIT-.mws ): Cylidical Coe Rod > R:.5;Ac:3.4*R^;P:*3.4*R; > Tb:;Tif:; R :.5 Ac :.785 P :.34 Tb : Tif : Mateial coe (oeties i the table ): > :4;h:5; : 4 h : 5 > M:sqt(h*P//Ac);M^; Note: Hee, M is M : M m hp A c > :; : > ho:89.;c:39; ρ : 89. c : 39 > alha:/ho/c; α : > a:/sqt(alha); a : > s:tif(tb-tif)*(cosh(m*(-)) (h/m/)*sih(m*(-)))/(cosh(m*)(h/m/)*sih(m*)); s :.7377 cosh( ) sih( ) CASE < < M > w():*sqt(-m^)*cosh(sqt(-m^)*)h*sih(sqt(m^)*); w( ) : cosh( ) 5 sih( ) > lot(w(),..*m^,y-..5); thee ae o eigevales i M the iteval < < M

24 878 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9 CASE M < > w():*sqt(-m^)*cos(sqt(-m^)*)h*si(sqt(-m^)*); w( ) : cos( ) 5 si( ) > lot(w(),m^..*m^); thee ae ifiitly may eigevales fo M < Eigevales: > :: fo m fom to do y:fsolve(w(),m^*m..m^*(m)): if tye(y,float) the lambda[]:y: : fi od: > fo i to 4 do lambda[i] od; > N:-; N : > :'':i:'i':m:'m':y:'y'::'': Eigefctios: > Y[]():si(sqt(lambda[]-M^)*); Y ( ) : si ( ) > N[]:it(Y[]()^,..): Eamle of Geealized Foie Seies: > f():^; f( ) : > b[]:it(f()*y[](),..): > ():sm(b[]*y[]()/n[],..n): > lot({(),f()},..,coloblac); this eamle demostates the ability of obtaied eigefctios to aoimate fctios by the tcated geealized Foie seies f b y y b f y d

25 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, > theta[]:(tb-tif)*sqt(lambda[]-m^)/lambda[]*(-e(- lambda[]/a^*t)); 8 θ : ( t ) ( e ) Solto: > T(,t):Tifsm(theta[]*Y[]()/N[],..N): Soltio with elicitly witte steady state soltio: > TT(,t):ssm((Tb-Tif)*sqt(lambda[]-M^)/lambda[]*(-e(- lambda[]/a^*t))*y[]()/n[],..n): > aimate({s,tt(,t),t(,t)},..,t..36fames); T b steady state soltio s fo aimatio of the temeate ofile see the website T t. iitial coditio > T:sbs(t,TT(,t)):T:sbs(t6,TT(,t)): T3:sbs(t8,TT(,t)): > lot({s,tif,tb,t,t,t3},..,coloblac,aesboed); T b steady state soltio s t 3 mi Note that the soltio with the elicitly witte steady state at odces o Gibb's effect t mi T t.

26 88 Chate IX The Itegal Tasfom Methods IX.5 Geealizatio of the Itegal Tasfom Method Jaay 3, 9