VIII.3 Method of Separation of Variables. Transient Initial-Boundary Value Problems

Transcription

1 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 65 VIII.3 Method of Sepaatio of Vaiables Tasiet Iitial-Bouday Value Poblems VIII.3. Heat equatio i Plae Wall -D 67 VIII.3. Heat Equatios i Catesia Coodiates -D ad 3-D 63 VIII.3.3 Heat equatio i Cylidical Coodiates 644 VIII.3.4 Heat equatio i Spheical Coodiates 654 VIII.3.5 Wave Equatio 665 VIII.3.6 Sigula Stum-iouville Poblem 67 VIII.3.7 Review Questios, Examples ad Execises 675

2 66 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7

3 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 67 VIII.3. HEAT EQUATION IN PANE WA -D Heat Equatio VIII.3.. BASIC CASE: Homogeeous equatio, Homogeeous Bouday Coditios [ u] x u ( x,t) u ( x) [ u] x u u α t u ( x,t) : x (,) Iitial coditio: ( x, ) u ( x) u, t > x Bouday coditios: [ u] x, t > (I, II o IIId kid) [ u] x, t > (I, II o IIId kid) ) Sepaatio of vaiables: u ( x,t) X ( x) T ( t) Bouday coditios: [ u] [ X] T( t) [ ] x x x [ u ] [ X ] T ( t ) [ ] x x x X X Equatio: X T µ X α T ) Stum-iouville Poblem: X µ X [ X] x [ X] x X ( ) x µ λ,,... 3) Equatio fo T : T αµ T + ( ) T αλ T T t e αλ t u x,t a X T 4) Solutio: ( ) t a X e αλ u x, u x a X Iitial coditio: ( ) ( ) a ( ) ( ) u x X x dx X ( ) x dx

4 68 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Example Neuma-Neuma Poblem u a u t u ( x,t) : x (,), t > Iitial coditio: ( x, ) u ( x) u Bouday coditios: u x x u x x t > (Neuma) t > (Neuma) (both boudaies ae isulated) Sepaatio of vaiables: u ( x,t) X ( x) T ( t) Bouday coditios: x x (,t) u (,t) u X X ( ) T ( t) ( ) T ( t) X ( ) X ( ) Solutio of SP: X µ X µ λ λ X π π λ X cos x,,... Solutio fo T: + ( ) + ( ) T αλ T T α T T t e αλ T t t u x,t a X T + a X T Solutio: ( ) t a + axe αλ a u ( x) dx ( ) π a u x cos x dx Solutio of IBVP: π π u( x,t ) u( x) dx u( x) cos x dx cos x e + α π t

5 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 69 Paticula case: ( ) u x + x 3 o C, s a 5 α m (steel),.m t 6 sec u ( x) t 6 t 3 t 6 u u ( x) dx a Commets: ) The solutio is i the fom of a ifiite seies. If the iitial tempeatue distibutio give by the fuctio u ( x) is itegable, the the Fouie seies is absolutely coveget ad the fuctio u x, t satisfies the Heat Equatio ad iitial ad bouday coditios. ( ) Theefoe, it is a aalytical solutio of the give IBVP. ) With the icease of time, the solutio appoaches the steady state (the aveaged tempeatue i the slab ). Boudaies ae isulated, ad thee ae o heat souces. As a esult, o heat escapes ito the suoudigs. The divig foce tempeatue gadiet is diected towad the aeas with lowe tempeatue. Thee exists a pocess of edistibutio of heat eegy that poduces the uifom tempeatue i the slab. 3) Basic fuctios cosist of the poduct o-dimesioal time Fo π αt t ( π π α ) π u ( x,t) cos x e cos x e whee the cosie fuctio povides the spatial shape of the tempeatue pofile; ad the expoetial fuctio is esposible fo decay of the tempeatue pofile i time. 4) The ate of chage of tempeatue depeds o the themal diffusivity α. 5) Vey ofte, a -D Heat Equatio is teated as a model fo heat tasfe i a log vey thi od of costat coss-sectio whose suface, except fo the eds, is isulated agaist the flow of heat Although, it is fomally a coect model, the pactical applicatio of it is vey limited. But thee is aothe itepetatio of a -D model, which is moe eliable. Coside a 3-D wall with fiite dimesio i the x-diectio (withi x ad x ) ad elogated dimesios (may be ifiite) i y- ad z- diectios. If the coditios at the walls x ad x ae uifom, ad the iitial coditio is idepedet of vaiables y ad z, the the vaiatio of tempeatue i the y- ad z-diectios is egligible (o heat flux i these diectios) u u y z ad the heat equatio becomes -D u u α t It defies the vaiatio of tempeatue alog ay lie pepedicula to the wall.

6 6 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Example Diichlet-Robi Poblem u ( x) u u α t u ( x,t) : x (,), t > u ( x,t) [ ] x u + Hu Iitial coditio: u ( x,) u ( x) Bouday coditios: [ u] x (Diichlet) u( ) x u + Hu dx x (Robi) h H k Sepaatio of vaiables: u ( x,t) X ( x) T ( t) Bouday coditios : X µ X T αµ T x X ( ) T ( t) X ( ) X ( ) + HX ( ) x X ( ) T ( t) + HX ( ) T ( t) Solutio of Stum-iouville poblem: µ λ ( λ ) X si x,,... whee eigevalues λ cos λ + H si λx λ ae positive oots of the chaacteistic equatio: Gaphically, they ca be show as itesectios of the gaph of the fuctio with the λ -axis: λ λ Solutio fo T(t): With detemied eigevalues, the solutio fo T becomes: ( ) T t e αλ t Solutio: ( ) ( λ ) u x,t a si x e αλ t This solutio satisfies the heat equatio ad bouday coditios. We wat to defie coefficiets a i a such a way that the obtaied solutio satisfies also the iitial coditio at t : ( ) ( λ ) ( ) u x, a si x u x

7 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 6 I ou poblem, fuctios { X ( x) si( λx) } ae obtaied as eigefuctios of the Stum-iouville poblem fo the equatio X + λ X ; theefoe, the set of all eigefuctios is a complete system of fuctios othogoal with espect to the weight fuctio u x p. The, the last equatio is a expasio of the fuctio ( ) i a geealized Fouie seies ove the iteval (,) with coefficiets defied by a ( ) ( λ ) u x si x dx si ( λ ) x dx Solutio : The, the solutio of the iitial-bouday value poblem is give by u ( x) si( λ x) dx αλ t u ( x,t) si ( λ x) e si ( λ x) dx whee the squaed om of eigefuctios may be evaluated afte itegatio as si( λ) X si ( λx) dx 4λ Fially, the solutio is: u ( x) si( λ x) dx αλ t u ( x,t) si( λ x) e si( λ ) 4λ

8 6 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Maple: et, H 3 > estat; > with(plots):, u ( x) x( x), α.65 > :;H:3;A:.65; : H : 3 A :.65 Chaacteistic equatio: > w(x):x*cos(x*)+h*si(x*); > plot(w(x),x..); w( x ) : x cos( x) + 3 si( x ) Eigevalues: Eigefuctios: > :: fo m fom to 5 do z:fsolve(w(x),xm/..(m+)/): if type(z,float) the lambda[]:z: :+ fi od: > fo i to 5 do lambda[i] od; > N:-; > :'':i:'i': > X[]:si(lambda[]*x); N : 3 X : si( λ x ) Squaed-om: > NX[]:it(X[]^,x..); NX : cos( λ ) si( λ ) + λ λ Iitial coditio: > u(x):x*(-x)+; u( x ) : x ( x) + Fouie coefficiets: > a[]:simplify(it(u(x)*x[],x..)/nx[]); a : ( λ + + si( λ ) ) λ cos( λ ) cos( λ ) λ λ ( cos( λ ) si( λ ) + λ )

9 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 63 Solutio - Geealized Fouie seies: > u(x,t):sum(a[]*x[]*exp(-lambda[]^*t/a^),..n): > plot3d(u(x,t),x..,t..3,axesboxed,stylewiefame); u ( x,t) t > aimate({u(x),u(x,t)},x..,t..5,fames,axesboxed); u ( x,t) > u(x,):subs(t,u(x,t)): > u(x,):subs(t,u(x,t)): > u(x,5):subs(t5,u(x,t)): > u(x,):subs(t,u(x,t)): > u(x,):subs(t,u(x,t)): > plot({u(x),u(x,),u(x,),u(x,5),u(x,),u(x,)},x..); u ( x) t t u ( x,t) t 5 t t t

10 64 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 VIII.3.. GENERA CASE No-Homogeeous Equatio, No-Homogeeous Bouday Coditios u u + F ( x) α t u ( x,t), x (,), t > Iitial coditio: ( x, ) u ( x) u Bouday coditios: [ ] x [ u] g u g, t > (I, II o IIId kid), t > (I, II o IIId kid) x I Steady State Solutio Defiitio A time-idepedet fuctio which satisfies the heat equatio ad bouday coditios obtaied as u s ( x) lim u( x,t) t is called a steady state solutio Substitutio of a time-idepedet fuctio ito the heat equatio leads to the followig odiay diffeetial equatio: us + F ( x) us ( x ), x (,) subject to the bouday coditios of the same kid as fo PDE: [ u ] [ u ] g, t > (I, II o IIId kid) s x g, t > (I, II o IIId kid) s x Geeal solutio of ODE: ( ) ( ) u x F x dx dx + c x + c s Solutios of BVPs fo plae wall with uifom heat geeatio ae povided by the Table. II Tasiet Solutio: Defie the tasiet solutio by equatio: U ( x,t) u( x,t) u ( x) s the solutio of the oigial poblem is a sum of tasiet solutio ad steady state solutio: u ( x,t) U ( x,t) + u ( x) Substitute it ito the Heat Equatio: U us U + + F ( x) α t us + F x, it yields Sice ( ) U a U t s

11 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 65 We obtaied the equatio fo the ew ukow fuctio ( x,t) has homogeeous bouday coditios: x [ U] [ u] [ u ] g g x x s x x [ U] [ u] [ u ] g g x x s x U which As a esult, we educed the o-homogeeous poblem to a U x,t with homogeeous bouday homogeeous equatio fo ( ) coditios. Iitial coditio fo fuctio U ( x,t) : ( ) ( ) ( ) ( ) ( ) U x, u x, u x u x u x s s Solutio fo U(x,t) We coside the followig basic iitial bouday value poblem: U U α t U ( x,t), x (,), t > iitial coditio: U ( x,) u ( x) u ( x) s bouday coditios: [ U] x, t > [ U] x t > We aleady kow a solutio of this basic poblem obtaied by sepaatio of vaiables: U ( x,t ) t axe αλ whee coefficiets a ae the Fouie coefficiets detemied by the coespodig iitial coditio fo the fuctio U ( x,t) : a ( ) ( ) ( ) u x us x X x dx X ( ) x dx III Solutio of IBVP: Solutio of the oigial IBVP is a sum of steady state solutio ad tasiet solutio: u ( x,t ) u ( x) + U ( x,t) s ( ) u x+ axe αλ s t whee a ( ) ( ) ( ) u x us x X x dx X ( ) x dx

12 66 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Example 3 Diichlet-Diichlet poblem with a uifom heat geeatio: u u + F α t u ( x,t) : x (,), t > Iitial coditio: ( x, ) u ( x) u u,t Bouday coditios: ( ) u (,t) g g t > (Diichlet) t > (Diichlet) ) Steady State Solutio: et F cost, the itegatig the equatio twice, we come up with the followig solutio: us Fx + c us F x + cx + c Apply bouday coditios to detemie the costats of itegatio: x c g x F c g g + + g g F c + F ( ) g g F u x x + x g + + s Example: F, g, g, 3 5 us ( x) x + x+ ) Tasiet Poblem: U U α t.5.5 U ( x,t) : x (,), t > iitial coditio: U ( x,) u ( x) u ( x) s bouday coditios: U (,t) (Diichlet) U (,t) (Diichlet)

13 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 67 Solutio of this basic poblem (Diichlet-Diichlet) obtaied by sepaatio of vaiables: π π λ, X ( x) si x U ( x,t ) t axe αλ π x e a si α π t whee coefficiets a ae the Fouie coefficiets detemied by the coespodig iitial coditio fo the fuctio U ( x,t) : ( ) ( ) ( ) u x u x X x dx π a u ( x) u ( x) si x dx s s X ( x) dx 3) Solutio of IBVP: Retu to the oigial fuctio u ( x,t ) : π u ( x,t) U ( x,t) + us( x) us( x) + a si x e α π t The the solutio of the o-homogeeous heat equatio with ohomogeeous Diichlet bouday coditios becomes: α π t g g π π ( ) s( ) F F u( x,t ) x + + x + g + u x u x si x dx si x e Remak: I pactice, istead of the exact solutio defied by the ifiite seies, the tucated seies is used fo calculatio of the appoximate solutio. How may tems ae eeded i the tucated seies fo the accuate appoximatio? Compaiso of the exact solutio (which is also a tucated seies but with a vey lage umbe of tems, which we assume, povides a accuate esult) with the calculatio with a small umbe of tems i a tucated seies shows that the accuacy depeds o time: the futhe we poceed i time, the moe accuate becomes a appoximate solutio (why?). Fo uifom chaacteizatio of physical pocesses, the o-dimesioal paametes ae used i egieeig. I heat tasfe, o-dimesioal time is defied by the Fouie umbe: αt Fo whee α is the themal diffusivity. I egieeig heat tasfe aalysis, a 4 tem appoximatio is cosideed as a accuate appoximatio fo all values of the Fouie umbe. Fo simplicity, vey ofte eve a tem appoximatio is used, which is cosideed to be accuate fo Fo >. (eo i most cases does ot exceed %, ad this is a covetio i egieeig heat tasfe).

14 68 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Coside compaiso of the exact solutio ( tems) with ad 4 tems appoximatios. Results ae calculated fo: Fo. Fo.5 Fo. Fo.4 The lowest cuve is a steady state solutio. As ca be see fom the figue, fo Fo >., all esults coicide. Fo >. is geeally adopted [see Icope ad De Witt] as a coditio fo applicatio oe tem appoximatio. Fo >.

15 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 69

16 63 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 VIII.3. -D AND 3-D HEAT EQUATION Catesia Coodiates Geeal Poblem: t > u u u + + F ( x,y) x y α t u ( x, y,t) : ( x, y) (,) (,M ) [ u] f ( x) y Iitial Coditio: ( x, y, ) u ( x, y) ( x,y) [,] [,M ] Bouday Coditios: u x [ u] f3 ( y) x (,M ) x [ u] f4 ( y) x (,M ) y [ u] f ( x) y (,) y M [ u] f ( x) y M (,) y t > y t > x t > x t > u s. We ae lookig fo a steady state solutio which satisfies the diffeetial equatio:. Steady State Solutio Fid time-idepedet solutio ( x, y) u u + + F ( x,y) y s s ad the bouday coditios of the same type as i the geeal poblem x [ us] f3( y) x (,M ) x [ us] f4( y) x (,M ) y [ us] f( x) y (,) y M [ u ] f ( x) (,) s y M y t > y t > x t > x t > This is the BVP fo Poisso s Equatio fo which, i geeal, all bouday coditios ae o-homogeeous. The supepositio piciple should be used to educe the poblem to the set of supplemetal basic poblems (see VIII.3.4, p.597).. Tasiet Solutio (Basic Case) Itoduce the tasiet fuctio as U ( x, y,t) u( x, y,t) u ( x, y) It ca be veified that fuctio U satisfies homogeeous Heat Equatio + x y α U U U t with fou homogeeous bouday coditios (of the same type): ad the iitial coditio: x [ U] x (,M ) x [ U] x (,M ) y [ U] y (,) y M [ U] y M (,) s y t > y t > x t > x t > ( ) ( ) ( ) ( ) U x,y, u x,y u x,y U x,y s

17 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 63 Sepaatio of vaiables st stage: We assume that the fuctio ( x, y,t) two fuctios U ca be witte as a poduct of U ( x,y,t) Φ ( x,y) T ( t) whee Φ ( x,y) is the fuctio of space vaiables. Substitute it ito the Heat Equatio Φ Φ T + T ΦT y α Divide equatio by Φ T : Φ Φ + y T Φ α T o usig aplacia opeato Φ T Φ α T eft had side is a fuctio of space vaiables oly ad the ight had side is a fuctio of the time vaiable, theefoe, they have to be equal to a costat (sepaatio costat): Φ T β Φ α T Bouday coditios fo sepaated fuctios ae: [ ] [ Φ ] ( ) x x (,M ) [ ] [ Φ ] ( ) x x (,M ) [ U] [ Φ ] T( t) y y (,) [ ] [ Φ ] T ( ) (,) y M y M y y x x t > [ Φ ] x t > [ Φ ] x t > [ Φ ] y t > [ Φ ] y M Thee ae fou homogeeous bouday coditios fo the fuctio Φ. Fom the sepaated equatios, coside the equatio Helmholtz Equatio Φ βφ which has a stuctue of equatio of the eigevalue poblem fo diffeetial opeato. It is called the Helmholtz Equatio. The solutio of the Helmholtz Equatio subject to bouday coditios ca be easily obtaied by the eigefuctio expasio method. Sepaatio of vaiables d stage: Assume Φ ( x,y) X ( x) Y ( y) Substitute ito the Helmholtz Equatio XY X Y + XY ζ XY ( ) Divide by XY X Y + β X Y Sepaatio of vaiables i the bouday coditios yield: y (,M ) [ Φ ] X ( ) Y( y) x [ ] x y (,M ) [ Φ ] X ( ) Y( y) x [ ] x x (,) [ Φ ] X ( x) Y( ) y [ ] y x (,) [ Φ ] X ( x) Y( M) y M [ ] y M X X Y Y Note, that we have complete pais of homogeeous bouday coditios both fo X ad Y.

18 63 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Now, solve cosequetly the Stum-iouville poblems fo X ad Y : X Y + β µ X Y Equatio is sepaated. It yields fist SP: X µ X SP µ λ [ X] x [ X] x X ( ) x,,... The the secod equatio becomes: Y + β λ Y which i its tu is a sepaated equatio: Y β + λ η Y It yields the secod Stum-iouville Poblem: Y ηy SP η ν m [ Y] y [ Y] y M Y ( ) m y m,,... Equatio fo sepaatio costats yields: β λ ν + m β ( λ + νm) The equatio fo T becomes T β ( λ + νm) α T Which is the st ode odiay diffeetial equatio: T + α λ + ν T ( m) with the solutios: ( m) T t e α λ + ν t m ( ) Solutio of the Tasiet Poblem: Costuct the solutio i the fom of double ifiite seies (eigefuctio expasio): ( m) U x,y,t A X Y e α λ + ν t ( ) m m m Whee the coefficiets ( ) ( ) U x,y, U x,y A X Y A m ca be foud fom the iitial coditio m m m as the Fouie coefficiets of the double Geealized Fouie seies: A M ( ) ( ) ( ) U x, y X x Y y dxdy m m X Ym

19 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, Example: DDNN u u u + y α t u ( x, y,t) : ( x, y) (,) (,M ), t > Iitial Coditio: ( x, y, ) u ( x, y) Bouday Coditios: u [ u] f ( x) y x [ u] x (,M ) x [ u] f4 ( y) x (,M ) y y M u y y u y y M y t > (Diichlet) y t > (Diichlet) x (,) t > (Neuma) x (,) t > (Neuma). Steady State Solutio Fid time-idepedet solutio ( x, y) u u + y s s subject to the bouday coditios: u s : x [ us ] x (,M ) x [ us] f4( y) x (,M ) y u s y y y t > y t > x (,) t > us y M x (,) t > y y M This is the basic poblem fo aplace s Equatio whe, thee bouday coditios ae o-homogeeous. Sepaatio of vaiables: us ( x, y) XY x [ u ] X s ( ) x x [ u ] f ( y) y s x 4 u s y y Y ( ) us y M y Y ( M) y M Sepaated equatio: Y X µ Y X Fist, coside equatio fo Y (two coditios): Y µ Y Y ( ) Y ( M) µ λ SP λ Y π M π Y y cos y cos y M λ ( ) ( λ )

20 634 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 The equatios fo X : X ( ) X λ X X x c+ cx X ( x) c cosh( λx) + c sih( λx) Bouday coditio at x yields X( ) c + c c c ( ) X c + c c c The X x x ( ) π X ( x) sih x M Costuct the steady state solutio as π π us ( x,y) axy + axy ax + asih x cos y M M This solutio should satisfy the bouday coditio at x ( ) ( ) s 3 u,y f y π π a + a sih cos y M M f y with Which is a cosie Fouie seies expasio of ( ) M a f ( y) dy 3 M M π a f3( y) cos y dy π M M sih M The the steady state solutio becomes: us ( x,y ) 3( ) + 3( ) 3 : M M π π π f y dy x f y cos y dy sih x cos y M π M M M M sih M. Tasiet Solutio Itoduce the tasiet fuctio as U ( x, y,t) u( x, y,t) u ( x, y) Fuctio U satisfies homogeeous Heat Equatio + y α U U U t with fou homogeeous bouday coditios: x [ U] x (,M ) x [ U] x (,M ) y y M ad the iitial coditio: U y y U y y M s y t > y t > x (,) t > x (,) t > ( ) ( ) ( ) ( ) U x,y, u x,y u x,y U x,y s

21 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, Sepaatio of vaiables U XYT yields a sepaated equatio X Y T + β X Y α T with homogeeous bouday coditios: x [ U] x x [ U] x y U X ( ) X ( ) y y Y ( ) U y M Y ( M) y y M Solve cosequetly the Stum-iouville poblems fo X ad Y : X Y + β µ X Y X µ X [ X] x [ X] x DD µ λ π, λ,,,... π X x si λx si x ( ) ( ) The the secod equatio becomes: Y + β µ λ Y which i its tu is a sepaated equatio: Y β + λ η Y It yields the secod Stum-iouville Poblem: Y ηy [ ] y Y [ ] y M NN Y m η ν m m,,,... Y y ν ( ) mπ M ν ( ) Equatio fo sepaatio costats yields: β λ η ν + m β ( λ + νm) mπ Ym y cos y M The equatio fo T becomes T β ( λ + νm) α T Which is the st ode odiay diffeetial equatio: T + α λ + ν T ( m) with the solutios: ( m) T t e α λ + ν t m ( ) Solutio of the Tasiet Poblem: Costuct the solutio i the fom of double ifiite seies (eigefuctio expasio): αλ ( m) t t U x,y,t A XYe A XYe α λ + + ν ( ) m m m

22 636 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Whee the coefficiets ( ) ( ) A m ca be foud fom the iitial coditio: U x,y, U x,y A X Y + A X Y m m m U ( x,y) A X Y + A X Y m m m whee M A X U ( x, y) dy M M A X U ( x, y) Y ( y) dy m m M The M A U x, y X x dydx ( ) ( ) M M 4 A U x, y X x Y y dxdy ( ) ( ) ( ) m m M 3. Solutio of IBVP u ( x, y,t) U ( x, y,t) + u ( x, y) s u ( x, y,t) m A m π mπ si x cos y e M mπ mπ + ax + amsih x cos y m M M π m π + t M a whee coefficiets ae M si π x dydx M A [ g( x, y) us ( x, y) ] 4 M mπ M π A m [ g( x, y) us ( x, y) ] cos y si y dydx M a M M f ( y)dy M mπ f y cos y dy mπ M M sih M a m ( )

23 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 637

24 638 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 4. Maple Example: heat5d-.mws, M 4,. 5 α, f ( y), g( x, y) x( x ) + y( y M ) -D Heat Equatio Example DD-NN > estat; > with(plots): > :;M:4;alpha:.5; > f(y):; > plot(f(y),y..m,axesboxed); : M : 4 α :.5 f( y) : fuctio i o-homogeeous bouday coditio [ u] f ( y) x 4 > u(x,y):x*(x-)*y*(y-m); u ( x, y) : x ( x ) y ( y 4 ) > plot3d(u(x,y),x..,y..m,axesboxed); iitial tempeatue distibutio ( ) ( ) + ( ) u x,y x x y y M Steady State Solutio: > a[]:it(f(y),y..m)//m; a : > a[m]:/m*it(f(y)*cos(m*pi*y/m),y..m)/sih(m*pi*/m); si( m π) a m : m π sih m π > us[m](x,y):a[m]*sih(m*pi*x/m)*cos(m*pi*y/m): > us(x,y):a[]*x+sum(us[m](x,y),m..): > plot3d(us(x,y),x..,y..m,axesboxed,pojectio.9); steady state solutio u s ( x,y)

25 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, > plot3d({us(x,y),u(x,y)},x..,y..m,axesboxed,stylewiefame); iitial tempeatue distibutio u ( x, y) steady state solutio u s ( x,y) Tasiet Solutio: > U(x,y):u(x,y)-us(x,y); U (, ) x y : x ( x ) y ( y 4) > A[,]:*it(it(U(x,y),y..M)*si(*Pi*x/),x..)//M: > A[,m]:4*it(it(U(x,y)*cos(m*Pi*y/M),y..M)*si(*Pi*x/),x..)//M: > U[,](x,y,t):A[,]*si(*Pi*x/)*exp(-^/^*Pi^*t*alpha): > U[,m](x,y,t):A[,m]*si(*Pi*x/)*cos(m*Pi*y/M)*exp(- (m^/m^+^/^)*pi^*t*alpha): > U(x,y,t):sum(U[,](x,y,t),..)+sum(sum(U[,m](x,y,t),m..),..): > U(x,y,):subs(t,U(x,y,t)): Solutio of IBVP: > u(x,y,t):us(x,y)+u(x,y,t): > u(x,y,):subs(t,u(x,y,t)): > aimate3d(u(x,y,t),x..,y..m,t..3,fames,axesboxed); x

26 64 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, D HEMHOTZ EQUATION Coside the Helmholtz equatio which appeas i the sepaatio of z all bouday coditios ae homogeeous [ ] I,II, o III U S vaiables i the Basic IBVP fo the 3-D Heat Equatio: U U U U t + + x y z α ( x,y,z) (,) (,M ) (,K), t > K y Sepaatio of vaiables: M ( ) Φ ( x,y,z) T ( t) U x,y,z,t iitial coditio: [ U] U ( x,y,z) t x Sepaated equatio: Φ Φ Φ + + y z Φ T β α T Sepaated equatio yields the Helmholtz Equatio: Helmholtz Equatio Φ βφ which costitutes the eigevalue poblem fo diffeetial opeato. The solutio of the Helmholtz Equatio subject to bouday coditios ca be easily obtaied by the eigefuctio expasio method. Assume ( x,y,z) X ( x) Y ( y) Z ( z) Φ Substitute ito the Helmholtz Equatio XYZ X YZ + XY Z + XYZ β XYZ Divide by XYZ ( ) X Y Z + + β X Y Z Sepaatio of vaiables i the bouday coditios yields: [ Φ ] X ( ) Y( y) Z( z) x [ ] x [ Φ ] X ( ) Y( y) Z( z) x [ ] x [ Φ ] X ( x) Y( ) Z( z) y [ ] y M [ Φ ] X ( x) Y( M) Z( z) y M [ ] y M [ Φ ] X( xy ) ( y) Z ( ) z [ ] z K [ Φ ] X( xy ) ( y) Z( K) z K [ ] z K x x y y z z Note, that we have complete pais of homogeeous bouday coditios fo X, Y ad Z. X X Y Y Z Z Now, solve cosequetly the Stum-iouville poblems fo X ad Y : X Y Z + β µ X Y Z

27 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 64 Equatio is sepaated. It yields the fist Stum-iouville Poblem: X µ X SP [ X] x µ λ ( ) [ X] x X ( ) x The the equatio becomes: Y Z + β µ λ Y Z which i its tu is a sepaated equatio: Y Z + β + λ η Y Z It yields the secod Stum-iouville Poblem: Y ηy SP [ Y] y η ν m ( ) [ Y] y M Y ( ) m y The oe moe step poduces equatio Z + β + λ νm Z which also ca be sepaated Z β + λ + νm γ Z It yields the thid Stum-iouville Poblem: Z γ Z SP,,,... m,,,... [ Z] z γ ω k ( ) [ Z] z K Z ( ) k z The the secod pat of the last equatio becomes β + λ + νm ωk ad the costat of sepaatio is k,,,... ( m k ) β λ + ν + ω The the solutio of the Basic IBVP fo the Heat Equatio is: ( m k ) t U x,y,z,t B X x Y y Z z e α λ + ν + ω Solutio of Basic IBVP: ( ) mk ( ) m ( ) k ( ) m k whee the coefficiets B mk ca be foud fom the iitial coditio as the Fouie coefficiets of the tiple Geealized Fouie Seies: B KM ( ) ( ) ( ) ( ) U x, y, z X x Y y Z z dxdydz m k mk X Ym Zk

28 64 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Pofesso Gabiel Węcel (Istitute of Themal Techology, Silesia Uivesity of Techology, Gliwice, Polad) has visited ou class o Febuay, 3

29 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, Summay Tasiet Coductio THE HEAT EQUATION Catesia Coodiates u u u u F ( x,y,z) y z α t [ u] S f 3 ( x,y,z) (,) (,M ) (,K) t > [ u] t t u M y [ u] f ( x) y M [ u] f ( y) x 3 u+ F [ u] f ( y) x 4 [ u] f ( x) y x ( ) ( ) + ( ) u x,y,z,t U x,y,z,t u x,y,z s STEADY STATE PROBEM - PE TRANSIENTPROBEM - HE us us us F ( x,y,z) y z + + y z α U U U U t [ u ] s S f [ U] S [ ] t t U u u s M y [ u ] f ( x) s y M M y [ U] y M [ u ] f ( y) s x 3 s + u F [ u ] f ( y) s x 4 [ U] x U U α t [ u] x [ u ] f ( x) s y x [ U] y x TRANSIENT SOUTION: U x,y,z,t B ( m k ) X Y Z e α λ ν ω t ( ) m k mk m k B MK ( ) u u X Y Z dxdydz s m k mk X Ym Zk

30 644 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 VIII.3.3. HEAT EQUATION IN CYINDRICA COORDINATES VIII.3.3. SOID CYINDER BASIC CASE: Homogeeous Equatio ad Bouday Coditios u u u + α t (, ) u t : [ ),, t > Iitial coditio: u (,) u ( ) Bouday coditios: u (,t ) < t > bouded [ ] u t > (I, II o IIId kid) ) Sepaatio of vaiables: u (,t) R( ) T ( t) u (,t ) bouded R ( ) < [ u] [ R] T( t) [ R] bouded R R T + µ R R α T ) Stum-iouville Poblem: R R + µ (Fom SP, µ λ ) R R + + R R λ R Bessel Equatio of ode Eigevalues: 3) Equatio fo T : T αµ T µ λ,,... Eigefuctios: R( ) J( λ) Weight Fuctio: p( ) + ( ) T αλ T T t e αλ t 4) Solutio: u (,t) art aj ( λ ) e αλ t u, u a R Iitial coditio ( ) ( ) a ( ) ( ) u R d R ( ) d ( ) ( λ ) u J d J ( λ ) d

31 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, GENERA CASE: No-Homogeeous Equatio, No-Homogeeous Bouday Coditios u u u + + F ( ) α t u ( x,t) : ( ),, t > Iitial coditio: u (,) u ( ) Bouday coditios: u (,t ) t > bouded [ u] f t > (I, II o IIId kid) I Steady State Solutio Time-idepedet solutio u ( ) s Substitutio of a time-idepedet fuctio ito the heat equatio leads to the followig odiay diffeetial equatio: u s u ( ) s + + F us ( ), (, ) subject to the bouday coditios of the same kid as fo PDE: [ u s ] x t > bouded [ us] f, Geeal solutio of ODE: u s + F( ) t > (I, II o IIId kid) u s F ( ) us F ( ) d + c us c F ( ) d + ( ) us F ( ) d d + c l + c Fo bouded solutio, it is ecessaily c, theefoe the geeal steady state solutio i cicula domai is ( ) u F ( ) d d + c s Solutios of BVPs fo cicula domai with uifom heat geeatio ae povided by the Table. II Tasiet Solutio: Defie the tasiet solutio by equatio: ( ) ( ) ( ) U,t u,t u s the solutio of the oigial poblem is a sum of tasiet solutio ad steady state solutio: ( ) ( ) + ( ) u,t U,t u s

32 646 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Substitute it ito the Heat Equatio: U U u u U α t us u + + F x, it yields s s F ( ) Sice ( ) U U U + α t We obtaied the equatio fo the ew ukow fuctio U (,t ) which has homogeeous bouday coditio: [ U] [ u] [ u ] f f s As a esult, we educed the o-homogeeous poblem to a U,t with homogeeous bouday homogeeous equatio fo ( ) coditios. Iitial coditio fo fuctio U (,t ) : U (,) u (,) u ( ) u ( ) u ( ) s s Solutio fo U(,t) We coside the followig BASIC iitial bouday value poblem: U U U + α t U (,t ), ( ),, t > iitial coditio: U (,) u ( ) u ( ) s bouday coditios: U (,t ) < t > [ U ] t > We aleady kow a solutio of this basic poblem obtaied by sepaatio of vaiables: ( ) U x,t art aj ( λ ) t e αλ whee coefficiets a ae the Fouie coefficiets detemied by the coespodig iitial coditio fo the fuctio U ( x,t) : a ( ) ( ) ( ) u us R d R ( ) d ( ) ( ) ( λ ) u us J d J ( λ ) d III Solutio of IBVP: Solutio of the oigial IBVP is a sum of steady state solutio ad tasiet solutio: u (,t ) u ( ) + U (,t) s

33 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, VIII.3.3. HOOW CYINDER BASIC CASE: Homogeeous Equatio ad Bouday Coditios u u u + α t u ( x,t) : ( ),, t > Iitial coditio: u (,) u ( ) Bouday coditios: [ ] [ ] u t > (I, II o IIId kid) u t > (I, II o IIId kid) ) Sepaatio of vaiables: u (,t) R( x) T ( t) [ u] [ R] T( t) [ ] [ u] [ R] T( t) [ ] R R T + µ R R α T R R ) Stum-iouville Poblem: R R + µ ( µ λ SP) R R + + R R λ R Bessel Equatio of ode Eigevalues: µ λ,,... λ ae oots of chaacteistic eq Eigefuctios: R ( ) c J ( λ ) + c Y ( λ ),, 3) Equatio fo T : T αµ T + ( ) T αλ T T t e αλ t 4) Solutio: u (,t) art ar ( ) t e αλ u, u a R Iitial coditio: ( ) ( ) a ( ) ( ) u R d R ( ) d

34 648 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 GENERA CASE: No-Homogeeous Equatio, No-Homogeeous Bouday Coditios u u u + + F ( ) α t u ( x,t) : ( ),, t > Iitial coditio: u (,) u ( ) Bouday coditios: [ u] [ u] f f t > (I, II o IIId kid) t > (I, II o IIId kid) I Steady State Solutio Time-idepedet solutio u ( ) s Substitutio of a time-idepedet fuctio ito the heat equatio leads to the followig odiay diffeetial equatio: us us + + F ( ) us ( ), (, ) subject to the bouday coditios of the same kid as fo PDE: [ u ] [ u ] s s f, t > (I, II o IIId kid) f, t > (I, II o IIId kid) Geeal solutio of ODE: ( ) u F ( ) d d + c l + c s Coefficiets c,c have to be detemied fom bouday coditios. II Tasiet Solutio: Defie the tasiet solutio by equatio: ( ) ( ) ( ), u (,t) U (,t) + u ( ) U,t u,t u s Substitutio ito the Heat Equatio yields a equatio fo tasiet solutio: s U U U + α t fo the ew ukow fuctio U (,t ) which has two homogeeous bouday coditios: [ ] [ ] [ ] U u u f f s [ ] [ ] [ ] U u u f f ad iitial coditio: s ( ) ( ) ( ) ( ) ( ) U, u, u u u s s

35 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, Solutio fo U(,t) We coside the followig BASIC iitial bouday value poblem: U U U + α t U (,t ), ( ),, t > iitial coditio: U (,) u ( ) u ( ) bouday coditios: [ ] [ ] s U t > U t > We aleady kow a solutio of this basic poblem obtaied by sepaatio of vaiables: ( ) U x,t art ar( ) t e αλ whee coefficiets a ae the Fouie coefficiets detemied by the coespodig iitial coditio fo the fuctio U ( x,t) : a ( ) ( ) ( ) u us R d R ( ) d III Solutio of IBVP: Solutio of the oigial IBVP is a sum of steady state solutio ad tasiet solutio: u (,t ) u ( ) + U (,t) s

36 65 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 VIII HEAT EQUATION i Cylidical Coodiates: u (, θ,z,t) -D AND 3-D cases u u u u g u z k α t θ ) og cylide u z : u u u g u θ k α t (,θ) θ ) Shot cylide with agula symmety u θ : u u u g u z k α t z (,z) 3) Cylidical suface of fixed adius u : u u g u + + θ z k α t z θ ( θ,z)

37 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, D HEMHOTZ EQUATION Coside the Helmholtz equatio which appeas i the sepaatio of vaiables i the Basic IBVP fo the 3-D Heat Equatio: z all bouday coditios ae homogeeous [ ] I,II, o III U S θ z α U U U U U t z K (, θ,z) [, ) [,π] (,K), t > Sepaatio of vaiables: ( θ ) Φ (, θ,z) T ( t) U,,z,t Sepaated equatio: iitial coditio: [ U ] U (, θ,z ) t Φ Φ Φ Φ θ z T β Φ α T Sepaated equatio yields the Helmholtz Equatio: Helmholtz Equatio Φ βφ The solutio of the Helmholtz Equatio subject to bouday coditios ca be obtaied by the eigefuctio expasio method. Assume (,,z) R( ) ( ) Z( z) Φ θ Θ θ Substitute ito the Helmholtz Equatio R ΘZ + R ΘZ + RΘ Z + RΘZ βrθz Divide by RΘ Z R R Θ Z β R R Θ Z Sepaatio of vaiables i the bouday coditios yields: [ Φ] R( ) Θ( θ) Z( z) x [ ] [ Φ] R( ) Θ( θ) Z( ) z [ ] z K [ Φ] R( ) Θ( θ) Z( K) z K [ ] z K z z We still eed the bouded solutio R Z Z [ Φ] R ( ) Θ( θ) Z( z) < [ ] ad peiodic R < (, +,z) (, +,z) Θ( θ + π) Θ( θ) Φ θ π Φ θ π

38 65 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 st equatio Θ R R Z Θ R R Z β η Θ ηθ ( + ) ( ) Θ θ π Θ θ η ( ) Θ θ η Θ( θ) a cos ( θ) b si( θ) +,,... d equatio R R Z β,,,... R R Z + β µ R R Z R R Z R R + µ R R + µ R R R R ( µ ) ( ) ( µ ) R + R + R µ λ R + + R + + R R λ R [ R ] < [ R ] ( ) ( λ ) + ( λ ) R c J c Y,, [ R ] < c, ( ) ( λ ) R c J, [ R ] ( ) J λ λ m,,,... m ( ),,,...

39 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, d equatio Z β λ Z m Z Z λ + β γ m Z γ Z SP [ Z] z γ ω k ( ) [ Z] z K Z ( ) k z The the secod pat of the last equatio becomes λm + β ωk ad the costat of sepaatio is k,,,... ( m k ) β λ + ω The the solutio of the Basic IBVP fo the Heat Equatio is: ( m k ) t U,,z,t A cos B si J Z z e α λ + ω Solutio of Basic IBVP: ( θ ) mk ( θ) + mk ( θ) ( λm ) k ( ) m k whee the coefficiets A mk ad Bmk ca be foud fom the iitial coditio as the Fouie coefficiets of the tiple Geealized Fouie Seies: Kπ ( ) ( λ ) ( ) U x, y, z J Z z ddθdz m k mk Rm Ym Zk A A B KM ( ) ( ) ( ) ( ) U x, y, z X x Y y Z z dxdydz m k mk X Ym Zk KM ( ) ( ) ( ) ( ) U x, y, z X x Y y Z z dxdydz m k mk X Ym Zk,,... m ( ),,,... k ( ),,,...

40 654 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 VIII.3.4 SOID SPHERE u,h Coside heat coductio i the sphee with agula symmety: u u φ θ he o-statioay tempeatue field u (,t) depeds oly o the adial vaiable ad time vaiable t. Iitial-bouday value poblem: ( ) u q u + k α t [, ) t > iitial coditio: (, ) u ( ) [ ] u, bouday coditios: ( ) u k hu u t > u < t > whee u is the ambiet tempeatue ad h is a covective coefficiet. Rewite the bouday coditio i the stadad fom u h hu + u k k ) Supepositio of Steady State ad Tasiet Solutios: ( ) ( ) + ( ) u,t u U,t ) Steady State Solutio: s ( ) u q + k s bouday coditios: us h hu + u s k k t > u s < t > Geeal solutio: q( ) c us ( ) d d + c k Fo the solid sphee (bouded solutio at ): q( ) us ( ) d d c + k Fo uifom heat geeatio ( q cost ): q + + 6k 3h q u us ( ) ( )

41 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, ) Tasiet Solutio: U U α t iitial coditio: [, ) t > ( ) ( ) ( ) [ ] U, u u s bouday coditios:, U h + U k t > U < t > Itoduce the ew depedet vaiable (eductio to catesia case): ( ) U (,t) V,t Evaluate U U U U V V + V V + V V + V + V U V Equatio: U α t V V α t V α t V V α t which fomally is the -d Heat Equatio fo i the fiite iteval [, ), which equies two bouday coditios. The fist coditio at is obtaied diectly fom the equatio used fo a chage of vaiable: V U Diichlet Coside the secod bouday coditio at : U h + U k V hv + k

42 656 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 V V hv + k V h + V k V h + V k V h + HV H Robi k Iitial-bouday value poblem: V V α t ( ) ( ) ( ) ( ) V, U, u us V U D V h + HV H R k 4) Stum-iouville Poblem coespodig to the case of Diichlet- Robi bouday coditios (table SP): Eigevalues λ ae the positive oots of the equatio: λ cos λ + H si λ Eigefuctios X si λ X si 4λ ( λ ) Solutio (see Example, p.?): V (,t ) c si( λ ) c e αλ t ( ) ( ) ( λ ) u u si d s X 5) Solutio: u,t us V,t ( ) ( ) + ( )

43 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, ) Example (tukey-3.mws) Roastig of a tukey The tukey (W u 5 lb ) is assumed to be a sphee with the uifom iitial tempeatue o C. It is exposed to the covective eviomet at u with the covective o 5 C W coefficiet h. The tukey is cosideed to be doe whe its miimum tempeatue mk o eaches udoe 75 C (Stadad fo Califoia). Themophysical popeties of tukey meat used fo calculatio ae fom the Table (Sectio.3.4, p.78). > estat;with(plots): > alpha:.3e-6;ho:5;cp:354;k:.5; α :.3-6 > h:; > qdot:.; ρ : 5 cp : 354 k :.5 h : qdot :. > W:5.;VO:W/ho;:fsolve(4/3*Pi*^3VO,..); W : 5. > H[]:h/k-/; Specified Tempeatues: > uif:5;ud:75; > u:; VO : : H : uif : 5 ud : 75 u : Steady State Solutio: > us:qdot*(^-)/6/k+qdot*/3/h+uif; us : 5. Tasiet Solutio: chaacteistic equatio: > w(x):x*cos(x*)+h[]*si(x*); w( x ) : x cos( x ) si( x ) > plot(w(x),x..5); Eigevalues: > :: fo m fom to do y:fsolve(w(x),x*m..*(m+)): if type(y,float) the lambda[]:y: :+ fi od: > fo i to 4 do lambda[i] od; > N:-; N :

44 658 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Eigefuctios: > X[]:si(lambda[]*); X : si( λ ) > NX[]:/-si(*lambda[]*)/4/lambda[]; si( λ ) NX : λ > c[]:it(*(u-us)*x[],..)/nx[]; > u(,t):us+(/)*sum(c[]*x[]*exp(-alpha*lambda[]^*t),..n); u (, t) : 5. + ( si( ) e ( t ) si( ) e ( t ) si( ) e ( t ) si( ) e ( t ) si( ) e ( t ) si( ) e ( t ) si( ) e (.4549 t ) si( ) e ( t ) si( ) e ( t ) si( ) e ( t ) )/ Solutio: Symmetic Extesio: > u(,t):subs(-,u(,t)): > t:.5*6*:t:3*6*6:t:5*6*6:t3:7*6*6:t4:9*6*6: > z:subs(tt,u(,t)):z:subs(tt,u(,t)):z3:subs(tt3,u(,t)):z4:subs(tt4,u(,t)): > plot({u,us,ud,z,z,z3,z4},-..,coloblack,axesboxed); u (,t) t t 9 hous 5 mi Tempeatue at the cete: > uc:limit(u(,t),); uc : e ( t ) e ( t ) e ( t ) e ( t ) e ( t ) e ( t ) e ( t ) e ( t ) e ( t ) e (.4549 t )

45 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, > aimate({u(,t),uc,ud,u,us},-..,t..*36,fames,axesboxed); u doe > plot({uc,u,ud},t..*36,axesboxed,coloblack); u doe u (,t) t doe

46 66 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 VIII.3.4. Heat Equatio i Spheical Coodiates: u (, φθ,,t) u u u u g u + + si φ + + siθ φ φ si θ θ k α t ) Sphee with agula symmety u φ : u u u g u si θ θ k α t Example: Floatig ball (,θ ) θ φ ) Spheical suface of fixed adius u : u u g u si φ + + siθ φ φ si θ θ k α t ( φθ, ) θ φ Example: Diffusio of foeig mit cois i Face

47 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 66

48 66 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 VIII PDE i spheical coodiates Coside a BVP geeated by sepaatio of vaiables i a PDE i spheical coodiates. We will oly see what the Stum-iouville poblems ae i this case.. aplace s Equatio Recall the geeal fom of aplace s Equatio i spheical coodiates u,φ,θ, D : fo the fuctio ( ) u u u cosθ u u si θ φ siθ θ θ o with diffeetial opeatos witte i self-adjoit fom: () u u cosθ u u si θ φ siθ θ θ () sepaatio of vaiables Assume u, φ, θ R Φ φ Θ θ (3) ( ) ( ) ( ) ( ) Substitute ito equatio () R cosθ ΦΘ + R ΦΘ + RΦ Θ + RΦΘ + RΦΘ si θ siθ Multiply the equatio by RΦΘ R R Φ cosθ Θ Θ R R si θ Φ siθ Θ Θ Coside the axisymmetic case ( ): φ R R cos θ Θ Θ R R siθ Θ Θ Sepaate vaiables ad set both sides of the equatio equal to the same costat R R cosθ Θ Θ + µ R R siθ Θ Θ It yields two equatios: R R ) + µ R R which ca be ewitte i the fom R + R µ R (Eule-Cauchy equatio) o i the self-adjoit Stum-iouville fom ( ) ( µ ) R R (4) Solutios of this equatio ae sought i the fom R m.

49 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, ) cosθ Θ + Θ + µθ siθ Use chage of idepedet vaiable x cosθ, the d d dx d Θ Θ Θ siθ Θ dθ dx dθ dx Θ d d d d d d d Θ siθ Θ cosθ Θ siθ Θ dθ dθ dθ dx dx dθ dx d d d dx d d cosθ siθ cos si dx dx θ θ dx + dθ dx dx Substitute ito equatio dθ d Θ cosθ dθ cosθ + si θ siθ + µθ dx dx siθ dx Θ Θ Θ Θ d Θ dθ si θ cosθ + µθ dx dx ( x ) d Θ dθ x + µθ dx dx o i self-adjoit Stum-iouville fom: d dθ ( x ) dx dx This equatio is called egede s diffeetial equatio. It happes that its solutio is bouded oly if the sepaatio costat is a o-egative itege of the fom µθ (5) µ ( + ),,,... Its solutio cosists of egede polyomials P ( x ) (see Sec. 5.7).. Heat Equatio Coside the axisymmetic heat equatio fo u (,t ), spheical coodiates: u u u + a t u,t R T t Sepaatio of vaiables ( ) ( ) ( ) D, t > i Substitute ito equatio (6) R T + R T a RT divide by RT ad sepaate vaiables R R T + a µ R R T It yields two odiay diffeetial equatios. Equatio fo R is R + R µ R (7) which is a spheical Bessel equatio of zeo ode (see equatio (5) i Sec. 5.6 with, AAEM-II). Eigevalue poblem: R ( R ) µ R Its solutios ae give by spheical Bessel fuctios π J( ) j ( ) y ( ) π Y ( ) (6)

50 664 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Why we caot be completely satisfied with the method of sepaatio of vaiables? How about the time depedet bouday coditios, fo example?

51 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, VIII.3.5 WAVE EQUATION VIII D Catesia homogeeous equatio with homogeeous bouday coditios u a u t u ( x,t), (,) x, t > Iitial coditios: u( x, ) u ( x) u( x, ) u ( x) Bouday coditios: ( ) u(,t) t u,t, t > (Diichlet) + h u ( x,t), k t > (Robi) Deote H h k. Sepaatio of vaiables we assume that the fuctio ( x,t) u ca be epeseted as a poduct of two fuctios each of a sigle vaiable u x, y X x T t ( ) ( ) ( ) u X ( x) T ( t) ( x) T ( t) X ( x) T ( t) u X t ( x) T ( t) substitute ito equatio a X Afte sepaatio of vaiables, oe gets X T µ with a sepaatio costat µ X a T That yields two odiay diffeetial equatios: X µ X ad T µ a T. Stum-iouville poblem X µ X bouday coditios: x X ( ) T ( t) X ( ) X ( ) + H X ( ) x X ( ) T ( t) + H X ( ) T ( t) This Stum-iouville poblem has the followig solutio with µ : λ eigevalues λ ae positive oots of equatio λ cos λ + H si λ eigefuctios X ( x) si λ x The solutios of the secod diffeetial equatio T + λ a T ae T ( t) c cos λ at + c si λ at basic solutios: u ( x,t) X T si λ x( c cos λ at + c si λ at)

52 666 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 We ae lookig fo a solutio i the vecto space with the basis { ( x,t) } iitial coditios: u u u u : ( x,t) au( x,t) a si λ x( c cos λat + c si λat) ( x,t) si λ x( a c cos λ at + a c si λ at) ( x,t) si λ x( b cos λ at + d si λ at) t ( x, ) b si λ ax u ( x) u which is a geealized Fouie seies expasio of the fuctio f ( x) ove the iteval (,) with coefficiets b u ( x) si λxdx u ( x) si λ xdx si λ 4λ si λ xdx The deivative with espect to t of the assumed solutio is u( x,t) λ a si λx( b si λat + d cos λat) t The the secod iitial coditio yields u( x, ) t d λ a si λx u( x) t Agai, it ca be teated as a Fouie seies with coefficiets d λ a d u ( x) ( x) si λxdx u( x) si λ xdx u si λxdx si λ λ a 4λ si λ 4λ si λ xdx The the solutio of the iitial-bouday value poblem is: 3. Solutio u ( x,t) si λ x{ b cos λ at + d si λ at} si λ x u si λ 4λ ( x) si λ + xdx cos λat u ( x) si λxdx si λat λ a

53 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, Nomal modes of stig vibatio Solutio of the Wave Equatio is obtaied as a sum of tems of the fom u ( x,t) X T si λ x( c cos λ at + c si λ at) which we called the basic solutios, but as the cotibutos to the vibatio of stig, these fuctios ae kow as omal modes. I ou example, fo,,3,4,... they have the followig shape (see Maple file fo aimatio): > m:subs(,x[]*(b[]*cos(lambda[]*a*t)+d[]*si(lambda[]*a*t))): > aimate({m},x..,t..9); fudametal mode > m:subs(,x[]*(b[]*cos(lambda[]*a*t)+d[]*si(lambda[]*a*t))): > aimate({m},x..,t..9); fist ovetoe > m3:subs(3,x[]*(b[]*cos(lambda[]*a*t)+d[]*si(lambda[]*a*t))): > aimate({m3},x..,t..9); secod ovetoe > m4:subs(4,x[]*(b[]*cos(lambda[]*a*t)+d[]*si(lambda[]*a*t))): > aimate({m4},x..,t..9); thid ovetoe ovetoes stadig waves The fist of these omal modes is called the fudametal mode, othes ae called the fist ovetoe, the secod ovetoe, ad so o. The fequecy of oscillatio of the omal mode is iceased with its umbe ad is detemied by the coespodig eigevalue λ ad coefficiet a which has a physical sese of the speed of popagatio of the waves (speed of soud). Thee ae fixed poits i the vibatio of ovetoes. The whole motio of the stig is a supepositio of vibatio of all ovetoes with diffeet amplitude. The ivolvemet of diffeet modes i the vibatio of stig is detemied by iitial coditios. If fo epesetatio of the iitial shape of the stig at est, diffeet modes ae equied, the all of them will be peset i the udamped vibatio of the stig. But if the iitial shape of the stig is exactly oe of the ovetoes, the oly this mode will be peset the stig vibatio. This pheomeo is called stadig waves. Stadig waves do ot popagate, oly shik ad swell i the same shape.

54 668 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 VIII D pola coodiates Wave Equatio i pola coodiates with agula symmety u + u a u t u (,t), [, ], t > Iitial coditios: u (,) u ( ) u (,) u ( ) Bouday coditio: ( ) t u,t t > (Diichlet). Sepaatio of vaiables Assume u (, y) R( ) T ( t) the u u u R ( ) T ( t), R ( ) T ( t), R( ) T ( t). t Substitute ito the equatio R ( ) T ( t) + R ( ) T R( ) T ( t) a Afte sepaatio of vaiables (divisio by R ( t) T ( t) ), we eceive R R T + µ with a sepaatio costat µ R R a T it yields two odiay diffeetial equatios: R + R µ R bouday coditio T µ a T (,t) R( ) T ( t) ( ) u R. Solutio of Stum-iouville poblem Coside fist the bouday value poblem fom which we expect to obtai eigevalues ad eigefuctios fo costuctio of a fuctioal vecto space. Coside the equatio fo R ( ) fo which we have a homogeeous bouday coditio: R + R µ R R( ) Multiplicatio by yields R + R µ R Solutio of this equatio depeds o the fom of the sepaatio costat ( λ) + d K ( λ) di µ > µ λ R( ) d l + d µ µ dj ( λ) + dy ( λ) µ < µ λ See, how these solutios wee obtaied ad how the bouday coditio ca be satisfied: ) µ > Deote µ λ. The the equatio becomes R + R λ ( + ) R

55 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, This is a modified Bessel equatio of itege ode fo the idepedet vaiable x λ. Geeal solutio of this equatio is give by ( ) d I ( λ ) + d K ( ) R λ At the iteio poit of the membae, the solutio has to be fiite, theefoe, we equie the coefficiet K λ (which is ubouded at ) to be equal to zeo, d. Coside the bouday coditio: ( ) d I ( ) d R Theefoe, the case µ > leads to a tivial solutio. ) µ Equatio becomes d befoe fuctio ( ) R + R The ode of equatio ca be educed by a chage of idepedet vaiable R ~ R. The equatio fo R ~ is a fist ode liea diffeetial equatio R ~ + R ~ solutio of which is obtaied with the help of a itegatig facto ~ d l d R( ) d e d ( e ) d ( ) The the solutio of the oigial equatio is ~ d R ( ) R( ) d d d l + d To have a fiite solutio at poit, we must put d. The the bouday coditio leads to d, ad we ed up with a tivial solutio. 3) µ < Deote µ λ. The the equatio becomes ( ) R R + R + λ which we ca idetify as a Bessel equatio of ode. The geeal solutio of Bessel equatio of itege ode is give by R d J d Y λ ( ) ( λ ) + ( ) whee J ( λ ) ad ( ) Y λ ae, coespodigly, Bessel s fuctios of the st ad the d kid of zeo ode. Befoe cosideig the bouday coditio, we ca make oe obsevatio. The deflectio of membae aywhee i the domai [ ], is assumed to be fiite (moeove, the wave equatio is deived i the assumptio of small deflectio). The Bessel fuctio of the d kid Y λ appoaches whe goes to. I ou case, is the ( ) iteio poit of the membae. Theefoe, fo fuctio R ( ) to be bouded at, the coefficiet d should be equal to. The solutio of the Bessel equatio becomes R( ) dj ( λ) The bouday coditio implies d J λ ( ) ( ) R If we wat a o-tivial solutio, the d, ad we eceive ( λ ) J

56 67 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 The figue shows the gaph of the fuctio w( ) ( λ ) λ J with The oots of the equatio λ, λ,... ae the values of the paamete λ fo which the bouday value poblem has a o-tivial solutio (they ae called eigevalues). The coespodig solutios (eigefuctios) ae R ( ) J ( λ ) To detemie the weight fuctio, tasfom the Bessel equatio to the self-adjoit fom of the Stum-iouville poblem (.6 poblem #4). Fid the itegatig facto m( ) d e ad educe the equatio to the self-adjoit fom [ R ] λ R The, accodig to the Stum-iouville theoem, the set of fuctios R { J ( λ) },,... is a complete set of fuctios othogoal with espect to weight ove iteval [, ], e.g. R ( ) Rm ( ) d whe m solutio fo T The esult of a egative sepaatio costat physical sese of solutio fo T ( t) : µ λ agees with a µ at µ at ce + ce µ > T µ a T T( t ) c + ct µ c cos µ at + c si µ at µ < We expect a peiodic solutio fo a udamped vibatio of membae. Thee should be o costat tems eithe because bouday coditios ad the equatio ae homogeeous. Theefoe, oly the case of a egative sepaatio costat may be accepted fo ou poblem, µ λ (o positive eigevalues). The solutios T ( t) with detemied eigevalues ae T ( t) c cos λ at + c si λ at, 3. Basic solutios Fo basis fuctios we take the solutios of the wave equatio satisfyig bouday coditios, u (,t) J ( λ )( c cos λ at + c si λ at), We ae lookig fo solutio of the give i.b.v.p. i the vecto space spaed by this basis:,

57 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 67 u (,t) a J ( λ )( c cosλ at + c siλ at) J J,, ( λ )( a c cosλ at + a c siλ ),, at ( λ )( b cosλ at + b siλ ),, at We will choose the values of coefficiets i such a way that iitial coditios ae satisfied. 4. Iitial coditios Coside the fist iitial coditio ( ), ( λ ) ( ) u, b J u the coefficiets fo the geealized Fouie seies ae defied as ( ) ( λ ) u J d, J ( λ ) b d ( ) ( λ ) u J d, J b ( λ ) The secod coditio fo the deivative with espect to time (,t) u J ( λ )( b, λa si λat + b,λ a cos λat) t becomes u (,) b,λaj ( λ ) u ( ) t The coefficiets i this geealized Fouie expasio ae ( ) ( λ ) u J d,λa J ( λ ) b d ( ) ( λ ) u J d, λa J b ( λ ) 5. Solutio u (,t ) J ( λ ) ( b, cos λat + b, siλat) The solutio of the iitial-bouday value poblem fo the wave equatio is J( λ) ( ) ( ) ( ) ( ) aλ J ( λ ) u (,t) u J λ d cos λat + u J λ d si λat

58 67 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 VIII.3.6 SINGUAR STURM_IOUVIE PROBEM We studied a egula Stum-iouville Poblem i which the odiay diffeetial equatio is set i the fiite iteval ad both bouday coditios do ot vaish. I a sigula Stum-iouville poblem ot all of these coditios hold. Usually, the iteval is ot fiite, ad oe o both bouday coditios ae missig. Istead of bouday coditios, whe the solutio may ot exist at the boudaies, the eigefuctios should satisfy some limitig coditios. Oe of such equiemets ca be the followig: et y ad y be eigefuctios coespodig to two distict eigevalues λ ad λ, coespodigly. The they have to satisfy the followig coditio: lim p( x) [ y ( x) y ( x) y ( x) y ( x) ] lim p( x) [ y ( x) y ( x) y ( x) y ( x) ] + x x x x I the othe cases the absece of bouday coditios is because of the peiodical o cycled domai, whe we demad that the solutio should be cotiuous ad smooth x y y x y y ( ) ( x ) ad ( ) ( x ) I this case, it is still possible to have the othogoal set of solutios { ( x) } [ x, ]. x y o We will ot study the fomal appoach to solutio of such poblems, but athe discuss the pactical examples of its applicatio. Hee, we coside a iteestig example of a sigula SP i a cycled domai with o bouday coditios. Physical demostatio of this example ca be see o the ceilig of the hall of the Eyig Sciece Buildig. Example Coside vibatio of a thi closed ig stig of adius descibed i pola coodiates by deflectio ove the plae z u( θ,t), θ [, π ], t > The Wave Equatio educes to u u a cost θ t with iitial coditios u, u θ ( ) ( ) θ u ( θ, ) u ( θ ) t Thee ae o boudaies fo a closed stig, but athe a physical coditio fo a cotiuous ad smooth stig: u,t u π,t t > ( ) ( ) u u θ θ (,t) ( π,t) t > sepaatio of vaiables Assume u( θ,t) Θ ( θ ) T ( t) Substitute ito equatio Θ T a ΘT Sepaate vaiables Θ T a µ Θ T µ is a sepaatio costat Θ Coside µ Θ Θ µ Θ We aleady have expeiece with solutio of this special equatio fo egula Stum-iouville Poblems ad kow that i all cases except the case of both bouday coditios of Neuma type, oly a egative sepaatio costat,

59 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, µ λ, geeates eigevalues ad eigefuctios. Geeal solutio i this case is Θ ( θ ) c cos λθ + c si λθ This solutio suits ou poblem because it is peiodic. The values of λ which θ, π, ae satisfy peiodicity o the iteval [ ] π λ π Theefoe, solutios ae Θ ( θ ) c cos θ c si θ, +, Obviously, that fo all,,,... π is a peiod fo this solutio ad fo its deivative Θ θ c si θ c cos ( ) θ, +, With these values of the sepaatio costat, µ λ,,,,... coside the equatio fo T ( t) : T a T T + T a which also has a peiodic (i t ) geeal solutio T ( t) c3, cos t + c4, si t a a The peiodic solutio of the wave equatio ca be costucted i the fom of a ifiite seies: u ( θ,t) Θ ( θ ) T ( t) Θ ( θ ) T ( t) ( c cos θ + c si θ ) c cos t + c a a,, 3, 4, si t c,c3, cos θ cos t + c,c4, cos θ si t + c,c3, si θ cos t + c,c4, si θ si t a a a a b cos θ cos t + b a cos θ si t + b a si θ cos t + b a,, 3, 4, si θ si t whee coefficiets b ae ew abitay costats which ca be chose i such a way that this solutio will satisfy the iitial coditios. Coside the fist iitial coditio: t (, ) u ( θ ) θ a u ( b cos θ + b si θ ) b, 3, ( b cos θ + b si θ ), +, 3, which ca be teated as a stadad Fouie seies expasio of the fuctio θ,π. Theefoe, the coefficiets of this expasio ae u ( ) o the iteval [ ] b b b π π, u ( θ ) d π θ π, u ( θ ) d π cos θ θ π 3, u ( θ ) d si θ θ

60 674 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Fo the secod iitial coditio, diffeetiate the solutio fist with espect to t u θ t (,t) b, cos θ si t + b, cos θ cos t b3, si θ si t + b4, si θ cos t a a a a a a a a the apply the secod iitial coditio u ( θ, ) u ( θ ) b, cos θ + b t a b Whee the coefficiets ae detemied as π b, u ( θ ) dθ π π a b, u ( θ ) cos θdθ π b a π π 4, u ( θ ) d si θ θ a 4, si θ, + b, cos θ + b4, si a a Coefficiet b, ca be ay costat, it will ot ifluece the iitial speed of the stig, but ot to ifluece the iitial shape of the stig it has to be chose equal to zeo (othewise, iitially the stig will shifted by b, ad will ot be ceteed ove the plae z ): b, Theefoe, solutio of the poblem is give by the ifiite seies θ u( θ,t) b, + b cos θ cos t + b a cos θ si t + b a si θ cos t + b a,, 3, 4, si θ si t a whee coefficiets ae detemied accodig to abovemetioed fomulas. Coside paticula cases (Maple examples): ) isolated wave ) stadig waves

61 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, VIII.3.7 REVIEW QUESTIONS, EXAMPES AND EXERCISES

62 676 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 REVIEW QUESTIONS. What is the mai assumptio i the method of sepaatio of vaiables?. What is a sepaatio costat? 3. How does the Stum-iouville poblem appea i the method of sepaatio of vaiables? 4. What is the fom of the solutio of the IVBP i the method of sepaatio of vaiables? 5. How may tems ae eeded i the tucated ifiite seies fo accuate pesetatio of the solutio? 6. Give a example whe the solutio of the IBVP is give just by a oe tem tigoometic fuctio? How does it happe?

63 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, EXAMPES AND EXERCISES. 3 et D be a domai (ope coected set), ad let S D\D be the bouday of D (ecall Sectio VIII..). Show that if S is a poit of the bouday of D, the ay ope ball B (,R) with a adius R > icludes poits both fom D ad 3 \D, i.e. itesectio of B (,R) with the domai ad with the suoudigs is ot empty: ( ) 3 ad B (,R) ( \D) B,R D. Remak: this popety is usually used as the moe geeal defiitio of the bouday: If A is a abitay subset of (ot ecessaily domai), the x is called a bouday poit of A if fo ay adius R > : B( x,r) A ad B( x,r ) ( \ A). The the set A { x x is bouday poit of A} i. is called the bouday of A If S D\D is the bouday of domai D, the S is the bouday of S i geeal sese too. Examples of the bouday i geeal sese: a) (,] {,} b) { a} { a} (the bouday of a isulated poit is the poit itself) c) d) e) f) g) { }. a) Solve the Diichlet poblem fo the Heat Equatio: u u a u ( x,t) : x [,], t > t Iitial coditio: ( x, ) u ( x) u Bouday coditios: u (,t), t > (Diichlet) u (,t), t > (Diichlet) b) Sketch the gaph of solutio fo 3 ad a. ad iitial coditios: i) u ( x) ii) u ( x) x( x) iii) u ( x) si x c) Obseve the Maximum piciple fo the Diichlet poblem fo the Heat Equatio.

64 678 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 3. The Supepositio Piciple fo No-Homogeeous Heat Equatio with No-Homogeeous Bouday Coditio.: Heat Equatio: u + F ( x) a u t u ( x,t) : x (,), t > Iitial coditio: ( x, ) u ( x) u u,t Bouday coditios: ( ) u (,t) g, t > (Diichlet) g, t > (Neuma) Supplemetal poblems a) steady state solutio: us + F ( x) u g ( ) s u s ( ) f u s ( x) : x (,) b) tasiet solutio: U a U t U ( x, ) u ( x) us ( x) U (,t) t > U (,t) t > U ( x, t) : x (,), t > Fist supplemetal poblem is a BVP fo ODE. The secod supplemetal poblem is a IBVP poblem fo the homogeeous Heat Equatio with homogeeous bouday coditios. Show that u ( x,t) U ( x,t) u ( x) IBVP. Solve the poblem with + is a solutio of the o-homogeeous F ( x) 5, g, g 3ad u ( x) x( 4 x) s. Sketch the gaph of the solutio.

65 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, a) Solve the IBVP: u a u + F t ( x) u ( x,t), (,) iitial coditio: ( x, ) u ( x) u x, t > u,t bouday coditios: ( ) u (,t) b) Sketch the gaph of solutio with 4, a. 5, k., f t > (I) ( ) k + hu,t f t > (III) dx u ( x) x( x / ) + 5, f, f, F ( x) x 5. a) Solve the IBVP fo the Heat Equatio i the plae wall with distibuted heat geeatio: u u + F ( x) α t, ( x,t) u, (,) Iitial Coditio: ( x, ) u ( x) u x, t >, F( x) q x k Bouday Coditios: ( ) u,t t > u (,t) k h T u (,t) dx t > b) Sketch the gaph of solutio with.5, α.5, k 5, u ( x), T, h 5, q 6. a) Solve the Heat Equatio i the cylidical domai with agula symmety + a u u u t (,z) u : <, t > Bouday coditio: u (,t) t > Iitial coditio u (,) u ( ) b) Sketch the gaph of the solutio fo.5 a 3 ( ) u 6 +

66 68 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 7. a) Solve the Heat Equatio i the cylidical domai with agula symmety + a u u u t (,z) u, <, t > Bouday coditio: u (,t) f Iitial coditio u (,) u ( ) t > b) Display some ceativity i visualizatio of solutio fo.5 a 3 f 7 ( ) u 5 + c) Give some physical itepetatio of the poblem 8. Solve the IBVP fo the Heat Equatio i pola coodiates with agula symmety: + a u u u t u (,t), [ ),, t > Iitial coditios: u (,) u ( ) Bouday coditio: ( ) u,t k + hu (,t) f t > Ad sketch the gaph of solutio fo, a.5, k., h, f (hit: fist, fid the steady state solutio) u, ad ( ) ( ) 9. a) Solve the Heat Equatio i the aula domai with agula symmety (cylidical wall with uifom heat geeatio) u u q u + + k α t ( ) u,t : < <, t > Bouday coditio: u (,t) T u (,t) T Iitial coditio: u (,) u ( ) t > t > < < b) Display some ceativity i visualizatio of the solutio fo.5.6 k 5 α. T 5 T ( ) u q 5

67 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, EXAMPE Radiatio Iduced Themal Statificatio i Suface ayes of Stagat Wate Pofesso Raymod Viskata (o the left) Atalya, Tukey, Jue

68 68 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Radiatio Iduced Themal Statificatio i Suface ayes of Stagat Wate Based o papes: [] D.M.Side, R.Viskata Radiatio Iduced Themal Statificatio i Suface ayes of Stagat Wate, ASME Joual of Heat Tasfe, Feb 975, pp [] R.Viskata, J.S.Too Radiat Eegy Tasfe i Wates, Wate Resouces Reseach, Vol. 8, No.3, Jue 97, pp Itoductio: Ou Objective: The vetical tempeatue distibutio i a body of wate have impotat effects o chemical ad physical popeties, dissolved oxyge cotet, wate quality, aquatic life ad ecological balace as well as mixig pocesses i wate. Sola adiatio is ecogized as the piciple atual heat load i wates. Some ivestigatos have cosideed the adiatio to be absobed at the wate suface (i.e. opaque) ad othes teated the wate as beig semitaspaet but igoed the spectal atue of adiatio. Sice the ultaviolet (UV) ad ifaed (IR) pats of the icomig sola adiatio ae lagely absobed withi the fist cetimetes of the wate ad the visible pat (VI) peetates moe deeply ad caies sigificat eegy to depths, the modelig of wate as a gay medium is ope to questio ad eeds to be examied. I the woks of Raymod Viskata (Pudue Uivesity) ad cowokes, aalysis fo the time depedet themal statificatio of i suface layes of stagat wate by sola adiatio was developed. The tasiet tempeatue distibutio is obtaied by solvig the oe-dimesioal eegy equatio fo combied coductio ad adiatio eegy tasfe usig a fiite diffeece method. Expeimetally, sola heatig ( T 58K ) of wate is simulated usig tugste filamet lamps ( T chaacteistics. 35K ) i paabolic eflectos of kow spectal Aalytical ivestigatio of tasiet combied coductio-adiatio heat tasfe with two bad spectal model (VI-IR) of icidet adiatio. Spectal distibutio of emissive powe: T 58K T 35K q eflected q VI q IR x β exticto coefficiet of wate, m - Q( x) x

69 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, Model: Heat equatio: ( ) u Q x u + k α t u x, u x T Iitial coditio: ( ) ( ) u k heff u T + q IR dx Bouday coditios: ( ) Souce fuctio [ u] T x Q x q βe β (adiat eegy absoptio ate): ( ) x VI x Wate Popeties: Extictio coefficiet β 7 m kg Desity ρ 3 m Specific heat cp 48 J kg K Coductivity k.6 W m K Data: egth.38 m a. Solve the give IBVP: u ( x,t ) Tempeatue T Tif T 5 Visible iadiatio q 85 Ifaed iadiatio q 5 Efficiet covective coefficiet o C VI IR heff W m W m W m K u (.5m,3s ) o 8.3 C paticula value b. Sketch the gaph of the solutio fo t 5,,5,3,6,9 mi ad compae with Viskata s esults. c. You view o the poblem. How ca the accuacy of the model be impoved? What have you leaed fom this poblem?

70 684 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 Solutio: Heat equatio: ( ) u Q x u + k α t u x, u x T Iitial coditio: ( ) ( ) u k heff u T + q IR dx Bouday coditios: ( ) [ u] T x x x ( ) β VI Q x q e β u k + heff u heff T q dx + IR x u heff heff T + q IR + u dx k k x u + Hu f dx x h eff H, k f heff T + q k IR u u + F ( x) α t q x x (,) F( x) VI e k β β u + Hu dx x u T [ ] x f ( ) ( ) u x, u x T I Steady State Solutio: u s ( ) + F x x (,) u Hu s + s dx x us T x [ ] f us + F ( x) us q VIβ x F ( x) e β k us q VIβ β x e dx k us q VI β β x e d β x k β ( ) us q VI e β x + c k q VI β x u e dx+ c x+ c k s ( )

71 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, q u e + cx+ c kβ VI β x s Bouday coditios: x x q q k kβ VI βx VI βx e c H e cx c f q q VI + c + H + c f k kβ q VI H c + Hc + + f k β VI H c + Hc + qvi + heff T + qir k β heff kc + heff c + qvi + qir + heff T kβ q VI β e + c + c T kβ q VI β c + c e + T kβ f x heff T + q k IR I matix fom: h + q + q + heff T kβ eff VI IR k heff c kβ c q VI e β + T Use Came s Rule: k h + ( eff ) eff det k h heff + qvi + qir + heff T heff k det β q h β q VI e + T kβ kβ kβ c k heff ( k + heff det ) eff VI β qvi qir heff T heff e heff T h ( ) kβ c + eff VI β qvi qir heff T T heff e ( k heff ) q kβ c : q h h Tif + qi + q + k β T + k h q e ( β ) k β h

72 686 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 c heff k + qvi + qir + heff T k det β q h β q VI e + T + kβ kβ kβ k heff ( k+ heff det ) VI β eff k e kt qvi qir hefft c h β kβ qvi eff e β + kt + + q VI + q IR + h eff T ( k+ heff ) c : k q e ( β ) T + k β h Tif + qi + q + k h q h k β h q q h q q h T T h e e kt q q h T ( ) eff VI β VI β eff + VI + IR + eff eff VI + IR + eff q β x kβ kβ β kβ VI us e + x+ ( ) eff ( eff ) kβ k+ h k+ h II Tasiet Solutio: U ( x,t) u ( x,t) u ( x) s U U α t x (,) Supplemetal SP (RD): U + HU dx x R λ oots of chaacteistic equatio: [ U] x D X ( ) si λ x ( ) ( ) ( ) ( ) U x, u x u x U x s X Solutio: U ( x,t) a t X e αλ ( ) s( ) ( ) a u x u x X x dx X III Solutio: u ( x,t) u ( x) + U ( x,t) s

73 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, > U:subs(t,u(x,t)): > U:subs(t3,u(x,t)): > U:subs(t6,u(x,t)): > U3:subs(t9,u(x,t)): > U4:subs(t8,u(x,t)): 3 mi > U5:subs(t36,u(x,t)): > U6:subs(t54,u(x,t)): 9 mi Compaiso Cuet aalytical solutio Expeimat ad umeical solutio [Viskata]

74 688 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7. Fid the solutio of the IBVP fo the Wave Equatio u a u t u ( x,t), (,) x, t > iitial coditio: u ( x,) u ( x) u ( x,) u ( x) t bouday coditios: u (,t) t > (Diichlet) u (,t) t > (Diichlet) Sketch the gaph of solutio with, a. 5, ad a) u ( x)., u ( x) x ( x) 6π b) u ( x), u ( x) si x (obseve the pheomea called stadig waves). Fid the solutio of the IBVP fo the Wave Equatio u a u t u ( x,t), (,) x, t > iitial coditio: u ( x,) u ( x) u ( x,) u ( x) t bouday coditios: u (,t) + Hu (,t), t > (Robi) u (,t) t > (Diichlet) Sketch the gaph of solutio with 5, a., ad a) u ( x)., u ( x ) ( x) b) u ( x), u( x) X5( x) 3. a) Solve the IBVP: u a u + F t ( x) (eigefuctio) u ( x,t), (,) iitial coditio: ( x, ) u ( x) u x, t > u,t bouday coditios: ( ) u (,t) b) Sketch the gaph of solutio with 4, a. 5,. k, ( ) f t > (Diichlet) ( ) k + hu,t f t > (Robi) dx u x x( x / ) + 5, f, F x f, ( ) x

75 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, Fid the solutio fo vibatio of the aula membae with agula symmety: u u u + a t u (,t), (, ), t > Iitial coditios: u (,) u ( ) u, u t ( ) ( ) Bouday coditio: u (,t) t > u (,t) t > Ad sketch the gaph of solutio fo, a.5, u ( )( ), ad ( ) u.

76 69 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 No-Classical IBVPs z 5. (Flow Betwee Two Plates) x T w y T w T T ρc T q ρc T + v + z k k k t p p x T Te x T < z T T w z T Tw t T T v T e Fid steady state solutio fo q. Sketch the gaph fo Te 8, Tw, v,., fluid is wate. 6. (Tasiet Coductio i Fi) h,t Tb T x T hp q ρcp T ( T T ) + ka k k t x T Tb x T T t T T c Fid tasiet state solutio fo q. cicula coppe fi ( D.5 ) Sketch the gaph fo Tb, T 5, T, T, h 5,.,

77 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, [Based o Nellis&Klei, p.37] Absoptio i a les Aalytical ivestigatio of tasiet combied coductio-adiatio heat tasfe with a gay spectal model of icidet adiatio. A les is used to focus the illumiatio adiatio that is equied to develop the esist i a lithogaphic maufactuig pocess The les is ot pefectly taspaet but athe absobs some of the illumiatio eegy that passes though it. covective-adiative suoudigs with T,h eff q icidet adiative flux x x.m dissipatio of the et adiative flux ito heat q q( x) ( x) x β exticto coefficiet of wate, m - tasmitted adiatio Model: Heat equatio: ( ) u Q x u + k α t u x, u x T Iitial coditio: ( ) ( ) u dx Bouday coditios: k h ( u T ) Dissipatio souce fuctio eff u k heff ( u T ) dx Q x q β e β (adiat eegy absoptio ate): ( ) x x x The es Popeties: Extictio coefficiet β m kg Desity ρ 5 3 m Specific heat cp 75 Coductivity k.5 Data: egth. m Tempeatue Tif T J kg K W m K W Icidet adiative flux q m Efficiet covective coefficiet heff o C W m K

78 69 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 a. Solutio of the give IBVP: u ( x,t ) b. Steady State Solutio: U( x ) c. Sketch the gaph.

79 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, Ivestigate the tempeatue field i the log colum of squae coss-sectio two adjacet sides of which ae o o themally isulated ad two othes ae maitaied at tempeatues C ad 5 C if iitially it was of uifom tempeatue T T o C. Sketch the tempeatue sufaces. T m u 9. Use sepaatio of vaiables fo solutio of IBVP fo log cylide with agula symmety. θ : u u u g u z k α t z (,z)

80 694 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7. Set up a mathematical model (choose a appopiate coodiate system ad dimesio of the poblem, wite the goveig equatio ad coespodig iitial ad bouday coditios) fo the followig egieeig models (do ot solve the poblem): a) A vey thi log wie dissipates eegy i the massive laye of the stagat media with the ate W W pe uit legth q. The media has a themal coductivity k m. Detemie the m K statioay tempeatue distibutio i the media. b) I the massive laye of homogeeous mateial (with themal popeties k, ρ,c p ) which was iitially at the uifom tempeatue T, a localized heat souce spotaeously stated to dissipate eegy with the ate q [ W ]. Detemie the developmet of the tempeatue field i the mateial. c) A vey log tee tuk of adius R i the foest is exposed to the suoudig ai (aveage wid m speed is v ), but the dese cow pevets the diect su adiatio of the tuk. Set up the s mathematical model descibig the tempeatue distibutio i the tee tuk duig the day. Coductivity i the tee depeds o diectio: it is much highe alog the tee tha i the adial diectio. W d) A wide esevoi of wate of metes deep is exposed to the sola iadiatio G m icidet at the agle θ. Peetatio of the sola adiative flux alog the path s is descibed by the s ambet-bee aw G( x) G cosθe κ, whee κ is the gay absoptio coefficiet of wate. m The the sola eegy dissipated i wate (adiative dissipatio souce o the divegece of dg Radiative flux) is detemied by ( ) ( x) W Q s dx 3. Set up the mathematical model m descibig the equilibium tempeatue field i the wate laye. e) Two opposite sides of the log colum ae isulated. Thee is a itesive codesatio of the wate steam o oe of the othe sides. The last side is exposed to the covective eviomet at tempeatue T ad covective coefficiet h W. Due to some chemical eactio thee is m K W poductio eegy i the colum with the volumetic ate q 3. Iitially, colum was at the m uifom tempeatue T. Descibe the tasiet tempeatue distibutio iside of the colum.

81 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, Staislaw Mazu ad Pe Eflo Mazu was a close collaboato with Baach at wów ad was a membe of the wów School of Mathematics, whee he paticipated i the mathematical activities at the Scottish Café. O 6 Novembe 936, he posed the "basis poblem" of detemiig whethe evey Baach space has a Schaude basis, with Mazu pomisig a "live goose" as a ewad: Thity-seve yeas late, a live goose was awaded by Mazu to Pe Eflo i a ceemoy that was boadcast thoughout Polad. vov i 9

82 696 Chapte VIII PDE VIII.3 Tasiet Iitial-Bouday Value Poblems Novembe 7, 7 AEGORY OF GEOMETRY Museum of ouve Ree Descates Uivesity, Pais