IX.4 The Hankel Transform

Transcription

1 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 83 I.4 T l Tsfom I.4. T l Tsfom Dfiiio Opiol Popis 84 I.4. Isos So 87 I.4.3 Eqio wi Isos Li So 88 I.4.4 Solio of PDE wi Appliio of Two Igl Tsfoms 86 (Foi l Tsfoms Diffsio of Lig i Smi-Ifii Sp I.4.5 Fii l Tsfom I.4.5. Cil Domi (FT- 8 I.4.5. Dvlopm of Vloiy Pofil i Pip Pss Gi I Al Domi (FT- 87 I Visos Fli Flow bw Two Cylis I Som iiol os o l Tsfom 84

2 84 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 I.4. ANKEL TRANSFORM ( is Bssl fio s of i of o. I is bo solio of Bssl Eqio ( y y y [ fo, ( x I pvios sios, w osi ppliio of Foi Lpl sfoms o solio of PDE s i Csi ooi sysm. Appliio of s sfoms limi ospoig ivivs fom iffil qio. I yliil ooi sysm, Lpli ils m W w o os igl sfom wi limis is m. Fomlly, s sfom b os fom woimsiol Foi sfom wi siio o pol oois ppliio of igl psio of Bssl fios [Db]. T sfom of fio, is fi s f of pol vibl [ { } ( ( f f f I is ll l Tsfom of o. T ivs l sfom of o is fi s { f } f f ( x ( x T oiios fo xis of l sfom simil o oiios of xis of Foi sfom, wi slly sisfi fo fios wi sib pysil mols i giig. T l sfom is ivs li opos. T pysil omi w w sppos o pply l sfom is ifii -imsiol pl wi pol,θ. T -ooi m of Lpli oois is obj fo l sfom. Simil o pvios mols, w lso ssm i is s fio is iviv v il yig ifiiy:, Opiol Popis To sy opiol popis, osi ppliio of l sfom of o o xpssio w iiol o Lpli so m fs o of ppli l sfom.

3 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 85 [ ] [ ] [ ] { } { } { } Tfo, o s o l Tsfoms I pil, zo o l sfom limis - m of Lpli o s o I f, w will s mosly zo o l sfom (ix i sfom fio is slly omi. Fom slf-joi fom of Bssl Eqio: y y wi : y y T fo y :

4 86 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 EAMPLE TRANSIENT EAT TRANSFER Coolig of bll-sp mp pofil Cosi xisymmi s of Eqio i yliil oois [, > b ifii ml pl b, (iiil oiio T o boy oiios b w ssm fio is iviv s il yig ifiiy: Pysilly i b s o bll sp mp pofil ifii pl moisl fom bo sis. is o log pl. Tml iffsiviy of pl is. Tsfom qio Apply zo o l sfom (, ( o qio o iiil oiio (s Tbl 7. Expoil fios 4b b (, b T is s o li ODE T solio is (, (, 4b 4b b b (, Ivs sfom Solio of poblm (s bl (, (,, 4b b b 4b 4 4b xp 4b 4 b

5 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 87 I.4. INSTANTENEOUS ENERGY SOURCE W osi volmi so S (, wi lss is g S, ( ( π δ δ W s g m g S, ( π δ δ W s g m g [ W s] g S, z z ( ( π δ δ δ g [ W s] g S, z z ( ( ( δ π δ δ i gy spoosly im pois [Ozisi, C, p.]. I xisymmi s (os o p o θ of yliil oois o p o z ooi ( sm so sg fo y pl ppil o z xis, is is yliil sf so wi b ps wi lp of Di l fio g W S(, δ ( δ ( π 3 m W s w g is so sg p i lg m g W s π is so sg p i m [ m ] vibl pps i omio s sl fo of sfomio o pol oois δ ( fis is of so m δ ( fis mom of im s Ig ov i im-sp omi (i pl ppil o z ooi, i sol yil sg of so (osvio of gy: π S (, π g θ δ ( δ ( θ π g π g π g π g π g π π π δ θ [ θ ] π δ ( δ ( ( θ θ I 3-imsiol s, so is il ig i pl z z fi by g S(, δ ( δ ( z z δ ( π A isos so is s fo miio of G fio fo o-omogos PDE. T so fo E qio is fi by S, w W m K is ml oiviy of mim W m is volmi gio. S 3

6 88 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 I.4.3 EAT EQUATION WIT INSTANTENEOUS LINE SOURCE Cosi xisymmi s of Eqio i yliil oois wi isos li so lo (li so i 3-D g δ π δ ( [, > Solio fo g S, ( π δ δ (, (iiil oiio T o boy oiios b w ssm fio is iviv s il yig ifiiy: Tsfom qio Apply zo o l sfom (, ( o qio iiil oiio g δ (, π ( ( is is s o li ODE g δ ( π T solio by viio of pm g π (, δ ( π π Ivs sfom g ( g ( (,, g ( ( π g π g π g 4 π (-.mws (, ( 4 ( ( ( ( 4 ( ( visi sp fio povis zo mp fo <, solio b wi i fom g 4π ( 4( >

7 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 89 Cyliil so 3 Cs g δ δ π g δ π g, δ π g π g π,, g π g π igio i [Ozisi, p.6, q(3-5] 4 I g π 4 I 4 g π Solio fo, 4 I 4 g π Tis is G fio fo Eqio i yliil oois.

8 8 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 -.mws l sfom solio of Eqio wi isos poi so > s; > wi(plos: > i:i(xp(-lmb^*(-/^* Bssl(,lmb**lmb,lmb..ifiiy: > (,:g*i//pi/; (, : 4 4 ( g ( π > g:;:;:;:; g : : : : > (,:sbs(-,(,: > (:sbs(.,(,:(:sbs(.5,(,: 3(:sbs(.,(,: > plo({(,(,3(}, ,xsbox,olobl;..5. > im((,, ,...,fms3; fo imio of mp pofil s wb si > im3([,,(,],...4,..*pi,.4..., fms,sylwifm,oosyliil; fo imio of mp pofil s wb si

9 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 8 Eqio wi isos yliil so wi > s; > g:;:;:;:;:; g : : : : : > (,:g/4/pi/(-*bssli(,^**//(-* xp(-(^^/4/(-*^; BsslI, (, : 4 π > (,:sbs(-,(,: > (:sbs(.,(,:(:sbs(.5,(,: 3(:sbs(.,(,: > plo({(,(,3(},-4..4,xsbox,olobl;..5. > wi(plos: > im((,,-5..5,...3,fms3; fo imio of mp pofil s wb si > im3([,,(,],...5,..*pi,.4..3, fms,sylwifm,oosyliil,olobl; fo imio of mp pofil s wb si --3.gif

10 8 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 I.4.5 T Wv Eqio (vibio of xisymmi ifii mmb [, i.. (, (, > Tsfom qio Apply zo o l sfom (, ( (, o qio iiil oiio (, ( is is o li ODE (, ( Gl solio ( si( os ( os( si Iiil oiios: (, (, Solio: os ( si( Ivs sfom: Solio of poblm is giv by ivs l sfom (, os( si(

11 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, Mpl xmpl (l-.mws: L mmb v iiilly bll sp : o A s T l sfom of is fio is s A Assm lso zo iiil vloiy of mmb T solio of is poblm is giv by followig igl os, [I.S.Gsi, I.M.Ryzi, Tbl of Igls #6.75.3] os b b 4 b b b 4 b iify: b T solio boms:, 4 4

12 84 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 l-.mws Wv qio - vibio of ifii mmb > s; > wi(plos: Iiil oiio: > (:/sq(^; ( : > yliplo([,,(],..6,..*pi; l sfom of iiil oiio: > b(lmb:simplify(fo(i(*bssl(,lmb**(,..ifiiy; b( : ( > v:; v : Solio - ivs l sfom: > (,:i(lmb*bssl(,lmb**b(lmb* os(v*lmb*,lmb..5; (, : 5 Bssl (, Tbl of igls [Gsi, Ryzi #6.75.3] > b:-;:-v*;:; b : - : : os( > (,:sq(sq((b^^-^^4*^*b^b^^-^/ sq((b^^-^^4*^*b^/sq(; (, : > (,:sbs(-,(,: > (:sbs(.,(,:(:sbs(.,(,: 3(:sbs(5.,(,:4(:sbs(.,(,: > plo({(,(,3(,4(},-3..3,xsbox,olobl;.. 5..

13 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 85 > im({(,,(,},-4..4,..,fms; fo imio s wb si > im3([,,(,],..4,..*pi,.., fms,sylwifm,oosyliil; fo imio s wb si > im3([,,(,],..4,..*pi,.., fms,sylpogi,oosyliil; fo imio s wb si

14 86 I.4.6 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 Solio of PDE wi ppliio of wo igl sfoms (Foi l sfoms Diffsio of lig ( i smi-ifii sp Cosi poblm of iffsio of lig i lf-sp z wi isos poi so Q S(, δ δ ( z z δ π Si poblm s gl symmy i xy-pl, i is sobl o osi i i yliil oois. I ply sig mim, iffsio ppoximio of lig popgio is sib by followig qio: isos poi so ϕ ϕ ϕ Q δ z π ( z δ δ T boy oiio sibs pil flio of lig fom sf pil sp of lig og sf ϕ ϕ z z Iiil oiio: ϕ w ϕ (, z, fly sibs iio fil 3( gµ iffsio offii µ sig offii g isoopy fo sp of lig Q so pow R R R sf fliviy (opil popy Dmi w yp of igl sfom o s: T Lpli of qio ils ivivs w... wo vibls: z, (Robi b. Foi s sfom [ [, l sfom Foi s sfom ϕ (, ϕ(, z, K(, z, z ( z, ϕ ( z,, K(, z ϕ, K (, z osz siz N ( π ( N l sfom ϕ (, ϕ (,,, (, ϕ (, ϕ,, N π

15 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 87 Apply igl sfoms o qio iiil oiio: Foi Tsfom δ N z z K z, N Q δ δ π ϕ ϕ ϕ l Tsfom ρ δ sfom qio N Q δ π ϕ ϕ ϕ solio by viio of pm ϕ τ τ τ τ δ π N Q N Q π 3 Ivs l Tsfom ϕ π N Q 4 N Q π N Q 4 4 π 4 Ivs Foi Tsfom ϕ π z K N Q 4, 4 π N z z N Q 4 si os 4 π z z Q 4 si os Solio z,, ϕ π z z Q 4 si os π z z Q 4 si os π z z z Q 4 si os os [ ] π z z z Q 4 si os os π π z z Q z 4 4 si os

16 88 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 SI-7b.mws Diffsio of Riio i smi-ifii sp fom isos poi so > s;wi(plos: Fly: > pi:q*//pi^/*xp(-*^/4//*(/*sq(pi*//*xp(- z^*/4//-^*i((os(omg*z/(omg^^*xp(- omg^**/,omg..5i((*omg*si(omg*z/ (omg^^*xp(omg^**/,omg..5; 5 φ Q / 4 π / 4 z os( z : si( z 5 ( π > :;:3;Q:;:;G:; : : 3 Q : : G : > f:sbs(.,pi/g: > f:sbs(-,f: > plo3({f,f},-..,z..,xsbox; isibio of fly i mim -9 Rflio fly sf z - lig xiig ly: > R(:sbs(z,f: > RR(:sbs(-,R(: > plo({rr(,r(},-..; -9

17 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 89 > siyplo({f,f},-..,z..,xsbox,sylpogi; - im > siyplo({f,f},-..,z..,sylpogi; -9 im > R:sbs(z,pi/G: > R:sbs(-,R: > im({r,r},-..,...,fms; fo imio of flio pofil s wb si

18 8 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 I.4.7 FINITE ANKEL TRANSFORM I.4.7. Cil Domi (FT- I is sio, w will os igl sfom, wi limis ivivs wi sp o pol ooi. S wi xiliy Sm-Liovill poblm o g igfios. Cosi wo pmi Bssl qios of o i fii il omi ( [, ] (BE Rwi BE qio i slf-joi fom ( Aoig o SLT, is qio wi omogos boy oiio w s ifiily my solios (igfios igvls oos of ospoig isi qio (bl CD. T s of igfios { ( } is oogol o [, ] wi wig fio b s fo Foi-l xpsio of fio : w ( ( ( If w g offiis i xpsio ( igl sfom pi of o b fi s ( Fii l Tsfom (FT Ivs Fii l Tsfom ( ( (? wi ivs sfom i fom of ifii sis. Bs fios ( solios of SLP, y solios of BE ( ( o wi i s ( ( sisfy o of omogos boy oiios: I Diil ( II Nm III Robi ( ( o ( (

19 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 8 Opiol popis of FT L som PDE il ms wi pol ooi wi objs fo limiio wi FT. Apply FT of o I [ ] [ ]

20 8 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 Fo pil yp of boy oiios, sl b spifi s followig: f I Diil ( ( f f ( ( f f o if o foml is s, ( ( f f f ( II Nm ( ( f ( f f o ( ( ( ( f III Robi ( ( ( ( f (

21 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 83 I.4.7. Dvlopm of vloiy pofil i pip pss gi Cosi visos fli wi is iiilly s isi log il yli, s i moio by spoosly impos iff bw psss s of pip. L G b xil pss gi, xil vloiy (, is sib by qio [Blo]: v G ρ [, ] wi iiil oiio (iiilly fli is s (, boy oiio, > Diil Igl sfom T fom of igl sfom is mi by omi (fii boy oiio (Diil, o of FT is mi by fis m i..s. of qio - o. Aoig o (Bssl Fios, sio, xmpl 3: Eigvls posiiv oos of ( Eigfios ( ( ( (, ( G S ( ρ ( G ρ G ρ Tsfom qio Opiol popy I of FT-I yils G ( ρ G ρ ( Solio by viio of pm G ( ( ρ 3 Ivs sfom Solio of IBVP: (, G ρ ( 3 ( 4 Sy s s G 4 ρ 3 ρ ( G ( 6 Mpl Exmpl (pip-.mws

22 84 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 pip-.mws Sig Flow i Pip (Blo, p.93 Solio wi Fii l Tsfom FT- > s;wi(plos: > :.;G:3;o:;:.5; :. G : 3 ρ : :.5 > w(x:bssl(,x*;plo(w(x,x..5; w( x : Bssl (,.5 x Eigvls: > lmb:y(..5; : y (.. 5, [ ] > :: fo m fom o o z:fsolv(w(x,xm*..(m*: if yp(z,flo lmb[]:z: : fi o: > fo i o o lmb[i] o; > N:-; N : 5 > :'':i:'i':x:'x':x:'z': Eigfios: > [](,:*G///o*Bssl(,lmb[]*/ lmb[]^3/bssl(,lmb[]**(-xp(-*lmb[]^*; (, : 6. Bssl (, 3 Bssl (,.5 Solio: > (,:sm([](,,..n: > :sbs(.,(,: > :sbs(.5,(,: > :sbs(,(,: > 3:sbs(.5,(,: Sy s solio (fom Blo: > f:g*(^-^/4//o; f : > plo({f,,,,3},..,olobl; (,..5 sy s fo imio of vloiy pofil vlopm s wb si

23 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 85 I Visos flow isi of oig yli Cosi moio of visos fli isi of il yli g by oio of yli o is xis. If moio of fli mis ply oiol xil ompo of vloiy is ql o zo vlopm of gl vloiy is sib by qio [ ] wi iiil oiio (iiilly fli is s (, boy oiio, > Diil os, Igl sfom T fom of igl sfom is mi by omi (fii boy oiio (Diil, o of FT is mi by so m i..s. of qio - o. Aoig o (Bssl Fios, sio, xmpl 3: Eigvls posiiv oos of ( Eigfios ( ( ( FT of (, (, ( Tsfom qio Opiol popy I of FT yils [ ( ] ( solio by viio of pm ( ( 3 Ivs sfom povis solio of poblm (, ( ( ( ( ( ( ( ( 4 Exmpl. [ ] m, 6 m s (w, 5 Sy s Sy s solio flly vlop flow os o p o im, qio is o Cy-El ODE, solio is s Sy s is obi fom sis solio s limi w ( ( s ( lim

24 86 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 pip-.mws Visos flow isi of oig yli Solio wi Fii l Tsfom FT- > s;wi(plos: > :.;omg:3;:.5; :. : 3 :.5 > w(x:bssl(,x*;plo(w(x,x..5; w( x : Bssl (,.5 x Eigvls: > lmb:y(..5; : y (.. 5, [ ] > :: fo m fom o 5 o z:fsolv(w(x,xm*5..(m*5: if yp(z,flo lmb[]:z: : fi o: > fo i o o lmb[i] o; > N:-; N : 4 > :'':i:'i':x:'x':x:'z': Eigfios: > [](,:*omg*bssl(,lmb[]*/lmb[]/ Bssl(,lmb[]**(xp(-*lmb[]^*-; (, 6 Bssl (, : Bssl (,.5 Solio: > (,:sm([](,,..n: > :sbs(.,(,: > :sbs(.5,(,: > :sbs(.,(,: > 3:sbs(,(,: Sy s solio (fom Blo: > f:omg*;f:-*omg; f : 3 f : 3. > plo({f,,,,3},..,olobl; sy s..5.5 fo imio of vloiy pofil vlopm s wb si

25 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 87 I Al Domi (FT- Axiliy Sm-Liovill Poblm. Cosi wo-pm Bssl qio of o i l omi, (BE wi omogos boy oiios: Aoig o SLT, is gl boy vl poblm s ifiily my solios (igfios ( Y (,, w igvls oos of ospoig isi qio (bl CD. T s of igfios { } is oogol o [, ] wi wig fio b s fo Foi-l xpsio of fio : w If w g offiis i xpsio igl sfom pi of o b fi s Fii l Tsfom (FT- Ivs Fii l Tsfom wi ivs sfom i fom of ifii sis.

26 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 88 Bs igfios solios of SLP, y solios of BE [ ] o wi i s [ ] sisfy o of ombiios of omogos boy oiios: I Diil II Nm III Robi T 9 possibl ombiios of s boy oiios. Opiol popis of FT- Cosi ppliio of FT- o followig xpssio Apply FT- of o [ ] [ ] [ ] [ ] [ ] Fo pil yp of boy oiios, is sl b spifi. T ompl solio of Sm-Liovill poblm i l omi is ps i bl SLP i l omi., osi ivio of opiol popis of FT- fo yps of boy oiios:

27 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 89 Robi-Robi f f [ ] [ ] f f ( Nm-Diil f f [ ] [ ] [ ] f f ( Diil-Nm f f [ ] f f (

28 83 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 f f ( Robi-Diil Nm-Robi ( f ( f f f ( ( ( ( ( ( Opiol popy: [ ] ( ( ( f f f f (

29 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 83 Nm- Nm ( ( ( f f [ ] [ ] ( ( f f ( N-N

30 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 83 I Visos fli flow bw wo ylis iiil oiio:, boy oiios: f Nm f Diil Igl sfom FT- of o posiiv oos of qio: Y Y Y Eigfios: Y Y Tsfom iiil oiio:, Tsfom qio Us qio ( : [ ] f f No:, is visosiy [ ] f f Q Noio: [ ] f f Q Solio by viio of pm, Q τ τ τ, Q τ τ (ssm Q Q os [ ], Q τ [ ] Q,, Q Q 3 Ivs sfom, Q Q 4 Sy s solio El-Cy qio x f x f s Q

31 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, FT--.mws Al omi : Nm-Diil boy oiios > s;wi(plos: > :omg*l*(-(x-l/(l-l; : > f:;f:omg*l; > omg:;omg:5;v:; > L:;L:3; > :; L x L L L f : f : L : : 5 v : L : L : 3 : Cisi qio (Tbl SLP i l omi: > :-x*bssl(,x*l/l*bssl(,x*l; : x Bssl (, x Bssl (, x > :-x*bssly(,x*l/l*bssly(,x*l; : x BsslY (, x BsslY (, x > :Bssl(,x*L; : Bssl (, 3 x > :BsslY(,x*L; : BsslY (, 3 x > w(x:*-*; w( x : x Bssl (, x Bssl (, x BsslY (, 3 x x BsslY (, x BsslY (, x Bssl (, 3 x > plo(w(x,x..; Eigvls: > lmb:y(..; : y (.., [ ] > :: fo m fom o 5 o y:fsolv(w(x,xm/..(m/: if yp(y,flo lmb[]:y: : fi o: > fo i o 4 o lmb[i] o;

32 834 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 > N:-; N : 4 >:'':i:'i':m:'m':y:'y':x:'x': Eigfios: > []:sbs(xlmb[],: > []:sbs(xlmb[],: > []:sbs(xlmb[],: > []:sbs(xlmb[],: > [](x:bssl(,lmb[]*x/[]- BsslY(,lmb[]*x/[]; Bssl (, x BsslY (, x ( x : Bssl (, 3 BsslY (, 3 Diviv of igfio, vls of igfio is iviv bois: > P[](x:iff([](x,x: > [](L:sbs(xL,[](x: > P[](L:sbs(xL,P[](x: Nom: > NN[]:i(x*[](x^,xL..L: Tsfom iiil oiio: > []:i(*[](x*x,xl..l: > Q[]:L*[](L*fL*P[](L*f: Foi-Bssl sis: > [](x,:(([]q[]/lmb[]^*xp(-v*lmb[]^*- Q[]/lmb[]^*[](x/NN[]: > (x,:sm([](x,,..n: Sy S Solio: > :(f-f*l/(l/l^/l::(f/lf/l^/(l/l^/l :s(x:/x*x: > im({(x,,s(x,},xl..l,..,xsbox,fms; sy s iiil vloiy fo imio of vloiy pofil s wb si

33 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, I Tsi sf og yliil wll (ml so V S E 5 impls boy flx, δ q α α < <, > (,, qδ, q δ f f, Fii l Tsfom of o (s Nm-Robi, Cp 3, p.38 Ivs Fii l Tsfom w igfios sq om,, ( x Y ( x ( x ( ( Y ( Y ( igvls posiiv oos of qio: ( Y ( Y ( Y ( ( ( 3 l sfom of qio (s opiol popy, p.93:

34 836 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 f ( f ( α α α q δ iiil oiio: 3 Apply Lpl sfom q su s U s U s s α α w q U s α s α 4 Apply Ivs Lpl fom q α ( α 5 Ivs l Tsfom solio of IBVP: q (, α ( α 6 Exmpl (pp of Y l, 9 ASME PVT.,.5,., α.6, , q Simpl wy o obi o solio fo vifiio.

35 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, Cyliil wll b ppoxim by pl wll. If vls of o big ifl of o boy is gligibl. Poblm b o followig: ( x, ( x, x fo x >, < < o oo big ( x,, x (, K q δ( Nm x Apply Lpl Tsfom: U( x,s su x,s x U(,s K q Nm x Solv: s x s x U x,s fo solio o b bo s x U x,s s U x,s x s x s s K U (,s K q x q K s U x,s s x q K s Apply ivs Lpl sfom o g solio: { } x, L U x,s q Sif solio o x K x 4 π Tmp pofil fo f piio: v x, ( x 4 q fo x K π, < < o oo big

36 838 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 Tis solio yils i smoo vs, solio is x, i povis vy goo piio fo im fo wi mp pofils o o so boy, ly b osi smi-ifii. Tis solio is goo fo ompiso wi solio i fii ly. Compiso of is solio wi solio by l sfom (48 ms i smmio: Fo.5.5 Solio lmos iil. I ms m bsolly ofi bo solios o. Fo so im <.5, osillios i l solio oiibl.!!! Tis solio.5 (o som o, fo xmpl. b s s iiil oiio fo Nmil (fii-iff solio of simpl poblm: α < <, >, ( 4α α q π, ( ( o impls!

37 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, I Simpl xmpl of Tsi sf og yliil wll (moifii Fb, 5 isl wll, qio i yliil oois: α Cosi followig iiil boy vl poblm: α < <, > Uifom iiil mp: (, Boy is moisl: f, Boy is xpos o oviv sf wi soigs mp: f, Fii l Tsfom of o (s Nm-Robi, Cp 3, p.38 Ivs Fii l Tsfom w igfios (Fii l of o T sq om of igfios is Eigvls posiiv oos of qio:,, ( x Y ( x ( x ( ( Y ( Y ( ( Y ( Y ( Y ( ( 3

38 84 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 T Fii l sfom of qio (s opiol popy, p.93: f ( f ( α α iiil oiio:, 3 Apply Lpl sfom o qio s α w Lpl sfom is fi by su s U s U ( s s α 4 Apply Ivs Lpl sfom α 5 Ivs l Tsfom solio of IBVP: U s (, α 6 Exmpl Coolig of sm (my fvoi foo o Tsim sq Mpl: SF-AD-6- Tsf Poblm N-R fii yli. Mws o T, C s 6 s 6 s 6 s 6 s

39 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 84 Boy vl poblms of oio Özışı, M. Ni So: Iiol Txboo Co., 968

40 84 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 oog Poblm T Eqio: α, < I Boilig II Coolig > T T s Iiil oiio: (, II (, II I (, I T Boy oiios: T s T W im is fo oog o b omplly oo? W im is fo oog o b ool og o o b b? 3 S mp fil of mil oss-sio s fio of im. 4 S mp of poi of mil oss-sio s fio of im.

41 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, I Som iiol os o ANKEL TRANSFORM Fb, 5 T l sfom of o of fio f of pol vibl [, { f } f f is fi s T ivs l sfom of o is fi s { f } f f Cllio of l sfoms is pfom by i ppliio of fiiio. T mi iffily will b igio. Tfo, spil bls of igls ivolvig Bssl fios sol b ppli. Igio b pfom wi lp of Mpl o y o omp sofw. B simpls is o v bl of l sfoms simil o Exmpls: Cll o l sfom of f Apply fiiio: f ( Cll ivs of o l sfom of f ( o os fio Apply fiiio of ivs sfom: f f. 3 Cll o l sfom of ( { ( } ( ( < < R f ( R > R f R R R R visi fio R R ( R

42 844 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 EAMPLE TRANSIENT EAT TRANSFER Coolig of yliil-sp mp pofil ( R ifii ml pl (, iiil mp isbio of i ifii ml pl o mp i si il of is R osi of il ( R R R.5,.,. (,.5 Cosi xisymmi s of Eqio i yliil oois [, > Iiil oiio (fo <R mp is T, fo >R, T < < R (, ( R > R T o boy oiios b w ssm fio is iviv s il yig ifiiy: Ipio: ifii pl moisl o bo sis. is o log pl. Tml iffsiviy of pl is. Tsfom qio Apply zo o l sfom (, ( o qio o iiil oiio (s Tbl 7. Expoil fios R R (, ( R T is s o li ODE T solio is (, ( R, ( R Ivs sfom Solio of poblm (,, R ( R R R (, R R ANKEL.mws Mpl mil igio: > :*R*vlf(I(Bssl(,*x*Bssl(,R*x*xp(-x^/^*,x..ifiiy;

43 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, Exmpl Tsi sf og yliil wll [poblm i Gzii] M, 5 - o fiis qio i yliil oois: α Cosi followig iiil boy vl poblm: α < <, > psib flx, q Uifom iiil mp: (, Boy is moisl: f, Boy, psib flx: q f q Fii l Tsfom of o (s Cp 3, p.38? Nm-Nm m? Ivs Fii l Tsfom,??? m (, w igfios (Fii l of o ( x Y ( x??? ( x ( ( Y ( Y ( T sq om of igfios is??? Eigvls posiiv oos of qio:??? ( Y ( Y ( Y ( ( 3

44 846 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 T Fii l sfom of qio (s opiol popy, p.93: f f ( N-N ( f q α q α α ( iiil oiio:, 3 Apply Lpl sfom o qio α U ( s α ( su s U s s Q s w Lpl sfom is fi by α U s Q s s α 4 Apply Ivs Lpl sfom (s ovolio om: α q α τ τ τ ( Exmpl: q ( b α ( α α ( τ q ( τ b τ α α ( τ q α ( τ ( τ ( b α ( τ ( b ( α α ( ( τ α α α b α 5 Ivs l Tsfom solio of IBVP: τ b α ( τ α τ τ ( ( τ m???,

45 6 Exmpl Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, Mpl: SF-AD-6- Tsf Poblm N-R fii yli. Mws o T, C s 6 s 6 s 6 s 6 s

46 848 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 4 Nm-Nm L L x Cosi BE i l omi ( x y x y xy, x ( L, wi omogos boy oiios: y x y x xl xl T gl solio is giv by ( x ( x Y ( x y L (Nm (Nm T iviv of gl solio (s i l iffil iiis y ( x ( x ( x Y ( x Y ( x x x x Sbsi io boy oiios: x ( L ( L Y ( L Y ( L L L L L x ( L ( L Y ( L Y ( L L L Do: L L L Y L Y L L L L L Y ( L Y ( L L T sysm fo offiis s followig mix fom: A ssy oiio fo sysm o v o-ivil solio is i yils isi qio fo vls of pm fo wi BVP s o-ivil solio:

47 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, Eqio fo igvls : ( L ( L L Y L Y L L Y L Y L L ( L ( L L T posiiv oos of is qio povi ifii s of igvls (o fo Nm boy oiios, is lso igvl fo Eigfios:. T fo mi igvls, offiis,, b fo fom o of qios of sysm (oos so o: O of offiis b s f pm, oos, Wi mi offiis, solios of BVP ( x (igfios v fom: y x fo y ( x ( x Y ( x,, ( x ( L ( L L Y ( x Y ( L Y ( L L T om of igfios is mi by igl L N, ( x L xy x, L L L x L L N, xy x x xx L L L Foi-Bssl sis: f x y x ( x y ( x w f fo L L L xy x f x x xf x x xf x x L L L L L N, xy ( x x xx L L L L xy x f x x xy x f x x L L L N, xy ( x x L y fo

48 85 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 I Comms bo ppoximio of Di fio fo V My 3, 5 Cosi boy oiio pp i Sio , δ q, q δ f C is oiio b milly ppoxim? I my opiio, sw is NOT! T Di fio δ is o lssil fio, bs i is o fi. T igoos fiiio of Di fio b o oly i ms of Toy of isibios. I ppli mmis, w s simplifi vsio of is fiiio: Dfiiio of Di fio δ : δ s isibio popy f δ ( f (, > fo y oios o (, fio f wi fii sppo Oly i ms of is fiiio w m y olsios bo ppoximio of δ. O fiiios s i pi, fo xmpl, δ δ ( NOT mmilly o y oly lp o v som i bo is fio. T my ppoximios of δ wi lso lp o s pysil mig of Di fio, b y o b s s mil ppoximio. Fo xmpl: sp fio wi ql o iy δ < > δ < < T slly yo s i is wi δ limδ B is ovg is o poi-wis bs limδ, owv δ is fi fo y.

49 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 85 I is oly o sy δ δ δ lim f f i ms of isibios: If yo s is ppoximio fo boy oiio, i will yil som sls. B y will o b mo if yo s, s i sol b wi pop ppoximio (lly, I v v i. If yo w o sig of poblm fo mil (fii-iff solio i b poply o i followig wy (ly isss bfo: (, Coos y smll vl (fo xmpl,. 3 T iiil-boy vl poblm fo mil solio os o ivolv Di fio: α < <, >, ( 4 q, K π iiil oiio (EACT!!! S ivio i p. Tis ppo wi smll og vl of smll og sp i o ps iiil pofil sol yil mil solio.

50 85 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 I.4.7. O-si Di-l wi b s i B.C. Fom viw of pp of Lpp [s viws] ii So, ipio of l Di fio is o osis (v io o oovsil mmil of Di fio: w δ s is wll-ow Di fio (wi s vl o if is zo lsw. Tog s psio is o fom mmil poi of viw... I o opiio, i is o o. Di fio is o if. Som fiiios ssig o Di fio if, b v. A mo sily spig, Di fio is o fio ll, b isibio, wi posssss isibio popy giv by qio ( δ s s. Moov, is popy sol b l ov y op ivl oiig poi s : δ s s fo y > (os o m ow smll. Tfo, isio of s Di fio o momm qio is o ppopi, bs Di fio is loosly ql zo osi of if, b if y is o il io omi of momm qio, wi is < y <. Somims, Di fio is ip s limi (o poi-by-by limi, wi b ll limi i isibio ss o w limi of sq of gl fios, fo xmpl, of sq of igl fios τ y y y y y y y, w ( y is visi i-sp fio. As b s, w, vl of fio zo τ. T isibio popy of is fio is τ ( y y fo y > (i pow. T, i limi i isibio ss, δ y limτ y (impls of i pow Tfo, fo Di l fio, op igbooo, wi ils pois fom bo sis of if is. T is, fiily, o s fo momm qio, wi is vli oly fo iio of omi, o vli boy osi of boy. So, spig of boy oiio y io omi wi lp of Di l is o possibl. I is ss, qios (,(3-5 is js symboli sipio of w o is yig o omplis. owv, o s ppo, i f, is o spig of boy oiio io omi. B is spig is pfom o wi lp of iiol Di fio, b wi lp of fio wi b fi s ψ ( y y

51 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, wi fo y > is fi fo ll ψ y. y y fo ll > ψ ( y ψ 4 ( y ψ 3 ( y ψ ( y T popy of is fio is simil o popy of Di fio, ψ ( y y ( y v is ql o iy owv, igio is oly ov iio si of if (i.. ov omi of govig qio. I limi (i isibio ss, o io impls fio of i pow ψ y limψ y wi sibs sp of i pow oly isi omi. Tis fio fomlly b obi by iffiio of fio ( y (6 i pp: φ ( y y y ψ ( y qio (7 b wi wi lp of ( y ψ s g ( y φ ( y C y C ψ ( y C y. φ fi by qio T, qio (7 b wi fo limiig s s g ( y ψ ( y C y

52 854 Cp I T Igl Tsfom Mos I.4 T l Tsfom Dmb 5, 7 Appliio of i s sig poi fo sq of f sps of obiig qios (8 (9 og wi boy oiios will yil limiig s of solio ily. Tfo, momm qio sol il fio ψ ( y, wi sibs spig oly io omi, is of Di δ, wi sibs spig io omi osi of omi. Sq of fios ( y s. ψ s, i f, i pp, sib oly ppo o limiig