ME 501A Seminar in Engineering Analysis Page 1

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1 Fobeius ethod pplied to Bessel s Equtio Octobe, 7 Fobeius ethod pplied to Bessel s Equtio L Cetto Mechicl Egieeig 5B Sei i Egieeig lsis Octobe, 7 Outlie Review idte Review lst lectue Powe seies solutios/fobeius Method ppl Fobeius ethod to Bessel s equtio Obtied idicil equtio lst wee Get fist solutio Diffeeces i secod solutio Defiitio of Bessel Fuctios Review Powe Seies Solutios Loo t followig equtio d poposed powe seies solutio Requies p, q d tht c be epded i powe seies bout = d d p q d d - d d - d d - Review Gettig the Solutios - p - q - Mipulte seies to get sigle sutio with coo powe of d coo liits Use substitutio of epoets to get coo epoets Reove tes fo sutios, givig idividul tes, plus coo su Review Gettig the Solutios II Result of ipultig su is seies tht hs fo c = C ol stisf this equtio if ll c = The c usull ivolve cobitios of the oigil tes This gives equtios betwee d coefficiets with subscipts -, -, etc. Iitil few coefficiets uow, used to tch boud coditios C get ll oigil i tes of these oigil coefficiets 5 Review Fobeius Method pplied to diffeetil equtio below Usul powe seies ethod ipplicble d b d c d d Solutio siil to pevious powe seies with = ecept fo fcto 6 ME 5 Sei i Egieeig lsis Pge

2 Fobeius ethod pplied to Bessel s Equtio Octobe, 7 ME 5 Sei i Egieeig lsis Pge 7 Review Fobeius Method II Diffeetite poposed solutio two ties Get powe seies fo b d c Substitute ito oigil equtio Set coefficiet of lowest te,, to zeo This gives idicil equtio, qudtic equtio with two oots fo, d Need two solutios but hve diffeet secod solutio depedig o d Se, diffe b itege, diffe b oitege 8 Review Fobeius Method III Fist d secod solutios d Double oot Fist solutio, ll cses Root diffeece ot itege l Roots diffe b itege be l 9 Bessel s Equtio ises i echicl d thel pobles i cicul geoeties The vlue of is ow pete Solve b Fobeius ethod d d d d d d d d Bessel s Equtio II Plug solutio d deivtives ito Bessel s equtio d ege Both Bessel s Equtio III Fil geet gets idicil equtio Idicil equtio = oots Solutio gives double oot if = Roots diffe b itege fo itege but ot fo o-itege Bessel s Equtio IV With =, we ust hve = With =, ll with odd vish Uow coefficiet fo iitil coditios o the diffeetil equtio Fo coefficiets of + to vish

3 Fobeius ethod pplied to Bessel s Equtio Octobe, 7 ME 5 Sei i Egieeig lsis Pge Bessel s Equtio V Get ew subscipt, = / = Test geel esult poposed below Get eve coefficiets,, i tes of Bessel s Equtio VI Copute / - fo geel equtio Result tches equtio fo lst cht Now hve geel esult fo fist oot of idicil equtio, = 5 Bessel s Equtio VII Fo itege =, ultipl b / Pic = / to give coveiet fuctios fo tbultio Use g fuctios to get siil esult fo o-itege 6 G Fuctios Fuctio geelizes fctoils to o-itege guets ppedi C dt t e t Defiitio log of + = + Fo itege =, + = = pplictio to Bessel coefficiets below 7 Bessel Fuctios Solutios use specific defiitio of = /[ +] fo tbles givig Substitute ito oigil solutio fo = Loo t itege d o-itege 8 Bessel Fuctios II Use fo itege vlues of Fo itege, + = Bessel fuctio, fist id, itege ode Fist few tes we chose J J Plots fo =,, d o et cht J

4 Fobeius ethod pplied to Bessel s Equtio Octobe, 7 J Bessel Fuctios III Bessel Fuctios of the Fist Kid fo Itege Odes J = J = fo = = = Bessel Fuctios IV Bc to Fobeius ethod fo secod solutios i thee cses = =, the double oot Itege =, oots diffe b itege, J - = - J No-itege, esiest cse, J d J - e two liel idepedet solutios Geel cse fo secod solutio J l Fo =, [,] = fist = Bessel Fuctios V Substitute poposed secod solutio ito oigil Bessel s equtio hee = - J l d dj d d d J d d [,] d l l J d d J dj l d d d d d Bessel Fuctios VI Result fo substitutio d d d d d J J dj l d d dj d l l J d d J l Rege to goup l tes Bessel Fuctios VII Coplete egeet, get deivtive d J dj l J J J d dj d d dj d d d Bessel Fuctios VIII Now substitute equtio fo deivtive ito geel seies equtio dj d dj d ME 5 Sei i Egieeig lsis Pge

5 Fobeius ethod pplied to Bessel s Equtio Octobe, 7 ME 5 Sei i Egieeig lsis Pge 5 5 Bessel Fuctios IX Substitute deivtive, ege sus 6 Bessel Fuctios X Equtio befoe choosig = o [,] [,] Double oot: = =, stts t j j j Rege lst two sus 7 Bessel Fuctios XI Result fo = with ew su tes We ust hve = fo te to vish Loo t coefficiet et Fist su hs ol eve powes of Loo t coefficiets fo odd 8 Bessel Fuctios XII Cop bsic equtio below Result fo, odd, is = - / Sice =, ll odd = Rewite bsic equtio fo eve powes of ol b settig = i secod su 9 Bessel Fuctios XIII Use usul powe-seies pplictio to ife geel equtio fo Get followig equtio fo Both sus ow hve ties ide Set coefficiets of = to zeo Bessel Fuctios XIV Do these two equtios stisf the pevious equtio fo i tes of -? Geel esult fo Rewite geel esult fo -

6 Fobeius ethod pplied to Bessel s Equtio Octobe, 7 Bessel Fuctios XV Gives coect esult fo Bessel Fuctios XVI So we ow hve secod solutio fo = J l We c use cobitio of two liel idepedet solutios fo secod solutio Defie Y = [ + l J ]/ = Eule costt which is liit s of the su + / + / + + / l Vlue of = Bessel Fuctios XVII This gives Y s follows Y J l Net step is gettig secod solutio fo itege Solutio poceeds i siil e I this cse we ust deteie if l te is equied i secod solutio See otes fo full detils s befoe defie Y Bessel Fuctios XVIII Y is defied s follows fo Y J l Geel solutio to Bessel s Equtio is = J + BY Plot of Y o et cht shows tht Y goes to ius ifiit s goes to zeo Bessel Fuctios XIX Bessel Fuctios XVIII Y Bessel Fuctios of the Secod Kid of Itege Ode = = = If we wt solutio fo = we cot use Y so geel solutio tht icludes = is = J Foll defie Y fo o-itege cos J J Y si I liit s ppoches itege, this defiitio ppoches Y ME 5 Sei i Egieeig lsis Pge 6

7 Fobeius ethod pplied to Bessel s Equtio Octobe, 7 Bessel Fuctio Su Bessel s equtio, d /d + d/d + - =, i pplictios e to pobles i dil geoeties. The geel solutio to Bessel s equtio is = C J + C Y whee C d C e costts tht e deteied b the boud coditios o the diffeetil equtio. Bessel s Equtio Su II J d Y : Bessel fuctios, ode, fist d secod id, espectivel. hve oscillto behvio foud i vious tbles d copute lib solutios t =, J = d J = s ppoches zeo, Y ppoches ius ifiit C tsfo soe equtios ito the fo of Bessel s equtio. 7 8 Clcultig Bessel Fuctios Ecel fuctios fo itege BESSELJ, coputes J BESSELY, coputes Y BESSELI, coputes I = i - J i BESSELK, coputes K = i - Y i Mtlb hs siil fuctios besselj, bessel, besseli, d bessel Ode of guets evesed u, Hdles o-itege Moe o Bessel Fuctios Fouls fo itegls d ecusio equtios Coputtiol ppoches G. N. Wtso, tetise o the Theo of Bessel Fuctios bowitz d Stegu, Hdboo of Mtheticl Fuctios, Ntiol Bueu of Stdds, 96 9 Fobeius Method Su The geel fo of the Fobeius ethod solutio is the ifiite seies = The geel solutio is diffeetited d substituted ito the oigil diffeetil equtio. Settig the coefficiets of ech powe of equl to zeo gives equtios tht c be solved fo d the i coefficiets Get coefficiets s i powe-seies Fobeius Method Su II Set coefficiet of = to get qudtic equtio fo idicil equtio Cses fo oots of idicil equtio the two oots e the se oots diffe b itege othe th zeo diffeet d diffeece is ot itege Fist solutio is lws = whee is lge idicil equtio oot ME 5 Sei i Egieeig lsis Pge 7

8 Fobeius ethod pplied to Bessel s Equtio Octobe, 7 Fobeius Method Su III Secod solutios deped o idicil equtio oots Roots diffeig b o-itege: = R , whee R is lge oot of idicil equtio Double oot: = l Roots diffeig b itege: = l whee be zeo Get i s i powe seies ethod Wht Hve We Leed? Powe seies ethod d Fobeius ethod used to solve soe equtios pplictio il i theo Give lticl solutio You ow solutio to Bessel s equtio = J + BY Pte give i equtio d B fit boud coditios B = to ppl solutio t = Wht C We Do With This? Bessel fuctios i Fouie seies Will use i ME 5B to get solutios to diffeetil equtio i dil geoeties Othe Bessel fuctios Hoewo poble o I = i - J i Copio fuctio K = i - Y i Solutios to siil equtios Tsfo diffeetil equtios ito Bessel s equtio 5 ME 5 Sei i Egieeig lsis Pge 8