ME 501A Seminar in Engineering Analysis Page 1

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1 St Ssts o Ordar Drtal Equatos Novbr 7 St Ssts o Ordar Drtal Equatos Larr Cartto Mcacal Er 5A Sar Er Aalss Novbr 7 Outl Mr Rsults Rvw last class Stablt o urcal solutos Stp sz varato or rror cotrol Multstp tods wt costat ad varabl stp sz Ssts o quatos Dto o st ssts Alorts or st ssts Fd sst s st Mr Probl O = = = Idcal quato = + = as tr cal roots o - so Plu tral P = A + B + C to ODE to d A = B = - ad C = 5 = 5 = = 5 = 5 Mr Probl O II = C =C + = C + = -5 + = -4 Boudar coo tat = s solvd to d C = 9 Soluto s 4 Mr Probl Two Solv us Laplac trasors [ = ad = -] Solv or Ys Rarra/us partal ractos or Ys 5 Mr Probl Two II Partal ractos rsults A = - B = 5 C = - D = so Ys quato s Fd vrs trasors vs t al rsult 6 6 ME 5A Sar Er Aalss Pa

2 St Ssts o Ordar Drtal Equatos Novbr 7 Mr Probl Tr Rarrat vs apls o Bssl s quato wt = ad =.5 d d d d For tr = = soluto s = AJ + BY ; or o-tr =.5 soluto s AJ.5 + BJ -.5 Ftt boudar coos vs rst rsult as.6j +.47Y ad scod as.8j.5.77j Ssts o Equatos A probl wt o or or ODEs o a ordr ca b rducd to a sst or rst ordr ODEs I a prvous lctur w rducd a sst o two scod ordr quatos to a sst o our rst ordr quatos I ts procss t su o t ordrs s costat Loo at prvous apl 8 Rducto o Ordr Eapl d d D varabls = d / ad 4 = d / T d / = d / ad d 4 / = d / Hav our sultaous rst-ordr ODEs d d 4 d d4 4 9 Sst o Equato Notato Wrt sst as d vctor quato d Idvdual d N copots d Eac a dpd o ad all d d 4 d d4 4 Solv Sultaous ODEs Appl sa alorts usd or sl ODEs Must appl ac stp ad substp to all quatos sst K s av cosstt ad valus rato o All valus ust b avalabl at t sa pot w coput t E.. Ru-Kutta w ust valuat or all quatos bor d Ru-Kutta or ODE Sst s vctor o dp varabls at = ad 4 ar vctors cota trdat Ru-Kutta rsults s a vctor cota t drvatvs = = = + / + / = + / + / 4 = = /6 ME 5A Sar Er Aalss Pa

3 St Ssts o Ordar Drtal Equatos Novbr 7 ODE Sst b RK4 d/d = - + z ad dz/d = z wt = ad z = - wt =. = [- + z] =.[- + -] = -. z = [ - z] =.[ - -] =. = [-+ / + z + z /] =.[ / /] = -.8 z = [+ / z + z /] =.[ + -./ /] =.8 ODE Sst b RK4 II = [-+ / + z + z /] =.[ / /] = -.8 z = [+ / z + z /] =.[ + -.8/ /] =.8 4 = [-+ + z + z ] =.[ ] = z = [+ z + z ] =.[ ] =.66 4 ODE Sst b RK4 III + = /6 = + [ ]/6 =.887 z + = z + z + z + z + 4z /6 = + [ ]/6 =.887 Cotu ts aso utl dsrd al valu s racd No dpdc or ts apl Nurcal Sotwar or ODEs Usuall wrtt to solv a sst o N quatos but wll wor or N = Usr as to cod a subrout or ucto to coput t arra Iput varabls ar ad ; s output So cods av o dsoal paratr arra to pass adoal orato ro a prora to t ucto tat coputs drvatvs 5 6 Drvatv Subrout Eapl Vsual Basc cod d or sst o d ODEs at rt s d d sow blow d d Sub sub as Doubl as Doubl _ as Doubl = - + Sqr + *Ep* = - * ^ = - * * Ed Sub 7 Drvatv Subrout Eapl Fortra cod or d sst o ODEs d at rt s sow d d blow d d subrout sub ralkind=8 : : = - + sqrt +*p* = - * ** = - * * d subrout sub 8 ME 5A Sar Er Aalss Pa

4 St Ssts o Ordar Drtal Equatos Novbr 7 Drvatv Subrout Eapl C++ cod or d sst o ODEs d at rt s sow d d blow d d vod subdoubl doubl [] doubl [] { [] = -[] + sqrt[] + []*p*; [] = - * [] * []; [] = - * [] * []; 9 St Ssts o Equatos Svral os Basc probl s tat tr ar svral lt or t costats valus t sst I o s atv ad lar atud copard to otrs ts wll st stablt Suc trs qucl drop to zro ad do ot act pscal soluto Howvr t orc sall or stablt Sl St Equato Noooous quato wt drt lt t scals d/d = = -a[ F] + df/d = [ F] - + F Soluto als o t two carts F = c + b: = [ b] - + c + b / = b/ - + c/ + b/ / = b/ - + c/ + b/ / - as ad / b/ + c/ as St Equato Soluto d/d = = -a[ F] + df/d soluto ro orula or a rst-ordr ODE d df a af d d d d C d ad C ad d C d F C St Equato Soluto II d df af d d afd df afd B parts udf = F a d F uf - Fdu C d C F C F F so F F Gar s Mtod Us plct alorts wt soluto b Nwto s tod Gral alort sow blow s lobal ordr o t tod Cocts: S tabl t sld Solv = b Nwto s tod ' 4 ME 5A Sar Er Aalss Pa 4

5 St Ssts o Ordar Drtal Equatos Novbr 7 ME 5A Sar Er Aalss Pa 5 5 Gar s Mtod Cocts / 4 - / / / / Nwto s Mtod Itrat to d suc tat = wrt Talor srs or stp ro old valu to w valu + O d d d d St + = ad solv or + Us ts to trat sc soluto s ot act 7 Solv d d d d Itrat to t + s trato d 8 Multvarabl Gar Mtod Iplct quato + Epad + Talor srs btw trato ad trato + or + N O ODE 9 Multvarabl Gar Mtod II W av a st o sultaous lar quatos to solv or + copots at ac trato N ODE Partal Drvatvs Nd partal drvatvs o ac wt rspct to ac d d d d d d 4

6 St Ssts o Ordar Drtal Equatos Novbr 7 Subrout or Partals Ts ca co drt ors dpd o arra allocato o cod O spl or s to us rpatd calls to t subrout or ac T d ad t valus o ad t arra ar passd to t rout ad t valus o t partal drvatvs o wt rspct to ac ar rturd a o dsoal arra p C++ Fucto or Partals vod pdrv doubl doubl [] t doubl p[] { == { p[] = -; p[] =.5 / sqrt [] ; p[] = p * ; ls == C++ Fucto or Partals II Group Wor { p[] = -4 * []; p[] = p[] = ; ls == { p[] = - * []; p[] = - * []; p[] = ; Start wor o t rst probl o t owor or Novbr 9 Solv t probl = -. wt = or our stps wt =. us t Adas-Moulto tod S t t sld or t alort as wll as t start valus ro t ourt-ordr Ru-Kutta Eact soluto s =. 4 Group Wor II Us Adas tod to solv = -. wt = or our stps wt =. act rror e E E ME 5A Sar Er Aalss Pa 6