adial diffsio eqaio Febay 4 9 Diffsio Eqaios i ylidical oodiaes ay aeo Mechaical Egieeig 5B Seia i Egieeig Aalysis Febay 4, 9 Olie eview las class Gadie ad covecio boday codiio Diffsio eqaio i adial coodiaes Solio by sepaaio of vaiables esl is fo of Bessel s eqaio eview Bessel fcios Eigefcio epasio i Bessel fcios eview Hoewo Poble Page 56 poble 5: fid, fo, f Sa wih sepaaio of vaiables solio [ si + cos [ si, e e cos Fo /, Fo /, π/ fo iege 3 eview Hoewo Poble II Geeal solio is s of all eigefcios π, Ae cos fo cosies π Geeal ohogoal f cos d elaioship fo A A π cos d Fial cosie esl has A π A f d A f cos d 4 T eview ovecio Poble Physical egio Hea Fl q - Syey odiio - Hea Fl q ξ Solio egio T odcio-covecio hea balace a q h T T τ T T Θ T T 5 eview Diesioless Poble Diffsio T Θ Θ eqaio τ ξ Iiial T, f f ξ T Θ ξ, codiio T T Syey boday Θ ξ codiio ξ ovecio boday codiio + h T T Θ h + Θξ ξ ξ Hoogeos 6 boday codiio ME 5B Egieeig Aalysis
adial diffsio eqaio Febay 4 9 Fcios 5 5 5-5 - -5 - -5 eview Fidig 4 6 8 4 6 8 / h /h ie oage Iesecios co 7 eview Iiial odiios sal fola fo b π Θ ξcos ξ Θ ξcos ξ cos si + cos ξ osa iiial epeae, T, gives Θ cos cos ξ si + si cos si + 8 eview Solio Geeal solio Θ ξcos ξ Θ ξ, τ e cos si + Solio fo T, T Θ Θ ξ, τ τ si e cos si τ cos ξ cos ξ + Need oo-fidig ehod o obai eigevales, h/ co 9 Diesioless Tepea eview Solio fo h/ a..9 a. a. a.5.8 a.5.7 a.75.6 a.5.4 a.5.3 a. a 3. a 4...3.4.5.6.7.8.9 / Diesioless Tepea eview Solio fo h/ a..9.8 a..7.6 a.35.5 a.5.4.3 a.75. a. a.5 a.5 a. a.5 a. a.3...3.4.5.6.7.8.9 eview Say eae S-ioville poble fo ozeo spaial boday codiios Defie, v, + w se ef i oigial eqaio Solve by sepaaio of vaiables Tie solio will be epoeial Apply boday codiios o deeie eigevales Also ges cosas ad which deeies fcios ha ae i solio / ME 5B Egieeig Aalysis
adial diffsio eqaio Febay 4 9 eview Say II Wie solio as s of all possible eigefcios wih idividal cosas se eigefcio epasio o ach iiial codiios If a solio fo, v, + w is sed he eigefcio epasio s be fo w Solio is s of all eigefcios wih cosas deeied fo achig iiial codiios ylidical Diffsio Eqaio Geeal diffsio eqaio fo hee diesios Oe-diesioal aesia + + y z adial eqaio ylidica l + + θ z coφ Sphee + + + si φ θ φ φ 3 4 adial Diffsio Eqaio Goves diffsio hea codcio i cylide fo ad, is epeae, species coceaio Iiial codiio, No boday codiio a ecep ha, is fiie / by syey Diffsiviy,, is aeial popey legh /ie 5 New Vaiable fo,, v, + v, saisfies diffsio eqaio v, ad v, is fiie Gives a S-ioville poble fo adial fcio Sice is a cosa,, is a solio o he oigial poble I saisfies he diffeeial eqaio ad he boday codiios v + v + 6 Sepaaio of Vaiables Asse v, PT [ P T P Divide by PT [ P T P T P T P 7 Solve ODEs o Ge v, Have epoeial ODEs i ie ad Bessel s eqaio fo adial fcio dt T T Ae d d dp This is fo of + P d d Bessel s eqaio Solio of Bessel s eqaio is P B + Y ad Y ae ode zeo Bessel fcios of fis ad secod id, especively 8 ME 5B Egieeig Aalysis 3
adial diffsio eqaio Febay 4 9 eview Bessel s Eqaio Gaa Fcios d y dy + + y d d Aises i echaical ad heal pobles i cicla geoeies The vale of is a ow paaee se powe seies solio echiqe ow as Fobeis ehod Have wo liealy idepede solios Fcio Γ geealizes facoials o o-iege ages Defiiio Γ e d Aalog of +! +! Γ + Γ Fo iege, Γ+! Γ sed fo coefficies of Bessel fcios wih oiege ode 9 Bessel Fcios, Fis Kid Sepaae epessios fo iege ad oiege vales of se fo iege vales of + + + +! Γ + +! +! Fis few es 4 + +!! +!! +! Plos fo,, ad 4 o e cha.9.8.7.6.5.4.3.. -. -. -.3 -.4 Bessel Fcio Plo Bessel Fcios of he Fis Kid fo Iege Odes 3 4 5 6 7 8 9 3 4 5 4 + Bessel Fcios, Secod Kid Y is defied as follows fo iege Y l + γ + π + + + +! +!! + No pese i eqaio fo Y! +! Geeal solio o Bessel s eqaio give by Bessel fcios of ode o y A + BY fo iege y A + B - fo oiege 3 Y.6.5.4.3.. -. -. -.3 -.4 -.5 -.6 -.7 -.8 -.9 - Bessel Fcio Plo Bessel Fcios of he Secod Kid of Iege Ode Y becoes ifiie as appoaches zeo 3 4 5 6 7 8 9 3 4 5 4 4 ME 5B Egieeig Aalysis 4
adial diffsio eqaio Febay 4 9 Bessel Eqaio Solios If we wa a solio fo we cao se Y so a geeal solio ha icldes is y A Foally defie Y fo o-iege Y cosπ siπ I lii as appoaches a iege, his defiiio appoaches Y Gives y A + BY fo ay Bessel Fcio Say Bessel s eqaio, d y/d + dy/d + - y, ai applicaios ae o pobles i adial geoeies. Physical poble gives vale fo The geeal solio o Bessel s eqaio is y + Y whee ad ae cosas ha ae deeied by he boday codiios o he diffeeial eqaio. 5 6 Bessel s Eqaio Say II ad Y : Bessel fcios, ode, fis ad secod id, especively. have oscillaoy behavio fod i vaios ables ad cope libay solios A, ad fo As appoaches zeo, Y appoaches is ifiiy a asfo soe eqaios io he fo of Bessel s eqaio. 7 adial Diffsio Solio Tasfo Bessel s eqaio whose solio is y A + BY Defie z / so y A z + BY z Tasfoed eqaio is d dy z + + z dz dz z adial diffsio eqaio has d dp + P Solio is P d d A + BY 8 Tasfoaio Deails d y dy + + y Bessel s eqaio d d Defie z / so ha z dy/d dy/dz /dy/d d y/d d y/dz / d y/d Sbsie io Bessel s eqaio ad divide eie eqaio by z d y dy z + z + z y z dz dz d y dy d dy z + + + + z z + z dz dz z dz dz z 9 Bessel as S-ioville opae asfoed Bessel s eqaio o S-ioville poble Bessel s eqaio S-ioville d dz d dz dy + z + z dz z dy z + dz [ q z + p z y Bessel is S-ioville eqaio wih z pz z, qz - /z ad Weigh fcio pz 3 ME 5B Egieeig Aalysis 5
adial diffsio eqaio Febay 4 9 adial Diffsio Solio II adial diffsio d dp + + P eqaio fo P d d Tasfoed d dy z + + z Bessel eqaio dz dz z Solio o secod geeal eqaio is y A z + BY z Solio o adial diffsio eqaio is A + BY is weigh fcio adial Diffsio Solio III Sa wih geeal solio P A + BY Solio applies i egio o Ms have P fiie a Sice Y - as we s have B fo P o be fiie a odiio P eqies Need solios,, of eqaio all solios o he zeos of 3 3.9.8.7.6.5.4.3.. -. -. -.3 -.4 Zeos of Bessel Fcios Bessel Fcios of he Fis Kid fo Iege Odes If, is a zeo of. Iiial poi whee is o a sefl zeo of 3 4 5 6 7 8 9 3 4 5 4 33 Bessel Fcio Zeos, 3 4 5.4483 5.58 8.65373.7953 4.939 3.837 7.559.7347 3.3369 6.4763 5.356 8.474.6984 4.79595 7.9598 ef: Abaowiz ad Seg, NBS AMS 55 Noe icease by abo π bewee sccessive zeos 34 Zeos of Bessel Fcios of he Fis Kid fo Iege Odes Zeos of Bessel Fcios of he Fis Kid fo Iege Odes.9.8.7.6.5.9.8.7.6.5.4.3. 4.4.3. 4.. -. -. -. -. -.3 -.3 -.4 -.4 4 7 3 3 4 4 3 4 5 6 7 8 9 3 4 3 4 5 5 3 5 6 8 9 5 35 36 ME 5B Egieeig Aalysis 6
adial diffsio eqaio Febay 4 9.9.8.7.6.5.4.3.. -. -. -.3 Zeos of 4 Bessel Fcios of he Fis Kid fo Iege Odes 4 Bessel Eigefcios, Diffee eigefcios ae all Bessel fcios of he sae ode, lie sies ad cosies we ca have ay diffee Bessel fcios Fo boh ids of fcios, he eigefcios ae give by he zeos siπ o cos[+π/ does o have fied ievals lie he sie ad cosie Fo adial solio, gives ; defie /, so -.4 3 4 5 6 7 8 9 3 4 5 4 4 34 37 38.8 Sie Eigefcios Sie π fo Diffee Vales..9.8 Bessel Eigefcios Bessel Fcio Hee so a.6.7.6.4.5 si p. -. 3.4.3... 4 -.4 -. -. -.6 -.3 -.4 -.8 -...3.4.5.6.7.8.9 39 -.5...3.4.5.6.7.8.9 4 Bessel Fcio Epasio Noalizaio Iegal Geeal eigefcio epasios Bessel eigefcios: a b p y a b p y d a f d f a y f a [ f d d p is weigh fcio 4 Iegal of d depeds o ad defiiio of Fo geeal esl is [ d [ + Fo his siplifies o [ d [ + Geeal esl ses Sve H fcio 4 ME 5B Egieeig Aalysis 7
adial diffsio eqaio Febay 4 9 ME 5B Egieeig Aalysis 8 43 Iegals o ope a Iegals wih Bessel fcios ay give coplicaed esls se sybolic i fcio of MATAB Soe siple cases ae possible sig eqaios o page 94 of Keyszig d sefl oly whe desied powe of aches he Bessel fcio ode 44 Eaple: f osc Bessel fcio epasio fo f i egio sig i case whee fo Hee we have a siple iegal wih so [ [ [ [ d d f a ξ ξ ξ ξ ξ 45 Eaple Epasio Bessel Fcio Epasio fo f..4.6.8.. 4. 6...3.4.5.6.7.8.9 Paial s of es es 3 es es 5 es 46 Bac o adial Diffsio Poble Have epoeial ODEs i ie ad Bessel s eqaio fo adial fcio, e + T d dt + P d dp d d Boday codiios: ad [, Y B Ae P T v + 47 Iiial odiio adial eqaio fo P is a S- ioville poble so we se eigefcio epasio fo iiial codiio egio is ad p is weigh fcio, + [ [ d d d 48 Eaple:, a osa [ [ [ d +, e f,
adial diffsio eqaio Febay 4 9..9.8 adial Diffsio a.8 a. a.6 a. a.4 a. a.6 a. a.4 a. [, - / -.7.6.5.4.3 a. a.3 a.5... a.4 a.5 a.6 a.8...3.4.5.6.7.8.9 / ME 5B Egieeig Aalysis 9