ME 501A Seminar in Engineering Analysis Page 1
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1 Powe Seies Solutios Foeius Metho Septee 6, 7 Powe Seies Solutios Foeius etho L Cetto Mehil Egieeig 5AB Sei i Egieeig Alsis Otoe 6, 7 Outlie Review lst wee Powe seies solutios Geel ppoh Applitio Foeius etho Bsi poess Applitio to Bessel s Eutio Review fo ite o Wees Review lst wee Sstes of iffeetil eutios eive fo phsil poles ivolvig iffeet, itetig pts Showe how to ovet two seo oe eutios ito oe fouth oe eutio Solve fouth oe eutio fo oe vile the use lge fo seo Fit iitil oitios o oth viles usig oth solutios Review Lst Wee II Solve sste of lie fist oe iffeetil eutios i ij j i t t j i, A t Solve i tes of eigevlues, l, eigeveto ti, X, fo A Review Lst Wee III Defie ew vile, s = X = Xs Tsfo oigil eutio s follows s s X X X AXs X s X p t t Tsfoe eutio is sl eutio whose solutio is ow it it it si e e pit Ci Cie i 5 Review Lst Wee IV Defiitios to ovet s i = C i e i t + i ito ti eutio s = EtC + E = I e E t t e t e t C C C C t e C 6 ME 5A Sei i Egieeig Alsis Pge
2 Powe Seies Solutios Foeius Metho Septee 6, 7 ME 5A Sei i Egieeig Alsis Pge 7 Review Lst Wee V Covet ti eutio fo s ito ti eutio fo usig = Xs Appl iitil oitio tht = t t =, whee E = I,: C = X Result: = XE [X ] + X Hoogeous = : = XEX 8 Review ODE Solutio Bsis A hoogeous th oe ODE hs sis of liel iepeet solutios / = hs the followig possile solutios: e, e, sih, osh ut ol two of these e liel iepeet Hve to fi oplete sis to e le to epeset ll possile iitil o ou oitios 9 Powe Seies Solutios Loo t followig iffeetil eutio popose powe seies solutio Reuies p, tht e epe i powe seies out = p Gettig the Solutios Diffeetite powe seies solutio sustitute it ito iffeetil eutio p Loo t siple eple with p = = = p Gettig the Solutios II I oe to stisf the powe seies eutio =, ll = Coie two sus ito oe with oo liits oo powes of Let = + i fist su Fist two tes i this su e zeo Rewite fist su so tht it hs se liits s seo su fte oppig fist two tes Mipultig Sutios Fist su ow hs se fto to liits s seo su
3 Powe Seies Solutios Foeius Metho Septee 6, 7 Mig Coeffiiets Vish Eh oeffiiet of vishes if = Gives eusio eutio fo Cotiue et slie Mig Coeffiiets Vish II Cotiue eusio eutio fo + Wt s futio of Popose geel eutio fo! with eve Che Eutio Wite geel eutio fo + the he tio + /!!!! Popose eutio gives se esult fo + / eive fo powe seies!! 5 Repet Poess fo o All o popotiol to Oigil solutio ow hs two seies Solutios e epete powe seies fo sie osie hose to fit iitil oitios!! 5! 5! 6 Su fo p Su Cotiues Wite the solutio fo s powe seies i uow oeffiiets Diffeetite the powe seies to get the eivtives euie i the iffeetil eutio 7 Get seies fo p,, if euie Sustitute ito iffeetil eutio Rewite the esultig eutio to goup tes with oo powes of. Set the oeffiiets of eh powe of eul to zeo givig eutio eltig eighoig vlues of Relte oeffiiets with highe susipts to those with lowe susipts. Iitil uow oeffiiets, e.g.,,, et., e fou fo iitil oitios 8 ME 5A Sei i Egieeig Alsis Pge
4 Powe Seies Solutios Foeius Metho Septee 6, 7 ME 5A Sei i Egieeig Alsis Pge 9 Su Colues Eie eutios eltig eighoig oeffiiets t to oti geel eutio fo eh i tes of the uow oeffiiets,, et. Sustitute the geel epessio fo ito the oigil powe seies fo This is the fil powe seies solutio Foeius Metho Applie to iffeetil eutio elow ou = whee / / e usul powe seies etho ipplile Solutio siil to pevious powe seies with = eept fo fto Foeius Metho II Diffeetite popose solutio two ties Get powe seies fo Sustitute ito oigil eutio Foeius Metho III Mipulte to get sigle sutio with oo powe of i eh te Foeius Metho IV Multipl esult oie ftos with + tes i sus Ep seies ultipl te te to get fist few tes i the seies fo the se whee = = A C B Foeius Metho V Coeffiiets of te ust vish + + = Do ot wt = This euies + + = C B A
5 Powe Seies Solutios Foeius Metho Septee 6, 7 ME 5A Sei i Egieeig Alsis Pge 5 5 Foeius Metho V Eutio + + = is ow s iiil eutio It is uti eutio givig two solutios fo ie Choose highe vlue of fo fist solutio Seo ODE solutio epes o vlues Doule oot, oots iffeig itege, oots iffeig oitege 6 Foeius Metho VI Fist seo solutios Doule oot A Fist solutio, ll ses Root iffeee ot itege l A Roots iffe itege e l A 7 Foeius Metho VII Ovell ppoh with this etho Covet ito powe seies if these e ot siple tes Fi iiil eutio oots Appl powe seies lsis to fi oeffiiets i eutio Bse o oots, eteie seo solutio Appl powe seies etho to fi A possil i oet eutio 8 Bessel s Eutio Aises i ehil thel poles i iul geoeties The vlue of is ow pete Solve s eple of Foeius etho 9 Bessel s Eutio II Plug solutio eivtives ito Bessel s eutio ege Both Bessel s Eutio III Fil geet gets iiil eutio Iiil eutio = oots = Solutio gives oule oot if = Roots iffe itege fo itege Roots o ot iffe itege fo oitege
6 Powe Seies Solutios Foeius Metho Septee 6, 7 Bessel s Eutio IV Cotiue et wee fte ite Get seies solutios fo Bessel futios fo thee ses Doule oot fo = Roots iffeig itege Noitege oots Fi two iffeet seies fo vlue of just lie fiig sie osie seies i powe seies solutio fo Review fo Mite Vetos, ties eteits Bsi opetios, ptiull ultiplitio Fi eteits ti iveses Vetos e liel epeet if i v i = with t lest oe i A sis set fo iesiol veto spe hs liel iepeet vetos tht epeset veto i the spe Guss eliitio poess fo solvig eutios eteies lie epeee Mite Review II Solutios to lie eutios A = Uiue if [A ] = A = N uows Ifiite solutios if [A ] = A is less th ue of uows No solutio if [A ] A Eigevlues eigevetos: A = DetA I = gives eigevlues Solve A I = fo opoets of eh eigeveto oe opoet it Mite Review III Digolize ti: Ʌ = X AX X is ti of eigevetos Ʌ is igol ti of eigevlues Wos ol if X hs ivese Speil ties Uit ti olus hve [* i ] j = ij Othogol ti olus hve [ i ] j = ij Heiti ti A* = A T Fo Heiti ti A = A T Mite Review IV Fistoe iffeetil eutios Seple fos, e.g. / = fg Geel lie eutio / + f = g hs solutio = e p [C + e p g] whee p = f Othe seple fos Solutios to / = f, eist ove egio out < i, /K whee, e is, oes K = f Uiue solutio if f/ is oue 5 Mite Review V Seooe iffeetil eutios with ostt oeffiiets: / + / + = : fi = gives hoogeous solutio, H Fo el, H = C e + C e Fo el = =, H = C +C e λ Fo ople oots, = Aos + Bsi, whee = β α/ = β α / Fo = H + P 6 ME 5A Sei i Egieeig Alsis Pge 6
7 Powe Seies Solutios Foeius Metho Septee 6, 7 Mite Review VI Fo ohoogeeous solutios fi solutio = H + P To get ptiul solutio, P Wite fo fo P, se o fo fo Sustitute postulte P with uow ostts ito ptiul eutio Eute oeffiiets of lie tes to fi uow ostts Use = H + P to fi ostts fo hoogeous solutio fo ou vlues Mite E Ope oo otes, iluig hoewo solutios Me ou ow otes to use fo e You e i toule if ou hve to use the oo o opeoo e M e useful to hve itegl tles Moe eit give fo showig how to oti solutio th fo poviig fil etils of lge o itheti 7 8 ME 5A Sei i Egieeig Alsis Pge 7
ME 501A Seminar in Engineering Analysis Page 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
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