Section 7 Ordinary Differential Equations
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1 Sect 7 Ordr Deretl Equts Deretl equts pl prtt rle egeerg sce te descrpts pscl pee re best rulted ters ter rtes cge. Deretl equts tt vlve e depedet vrble re clled rdr deretl equts; tse tt vlve re t e re clled prtl deretl equts. Deretl equts re clssed wt respect t ter rder. Secd-rder equts r eple clude secd dervtves. Fr eple te pst ss-sprg-dper sste s gve b te secdrder equt d d c dt dt Hger-rder deretl equts c be reduced t sste rst-rder deretl equts. Fr eple let v ddt d dvdt d dt s te secd-rder equt bve c be replced wt equvlet sste rst-rder deretl equts dv cv dt d v dt erere we wll rst cus te slut rst-rder deretl equts d lter dscuss te sluts sstes te. Csder te rst-rder rdr deretl equt d d Oe w t clculte te vlue t ew pst wuld be slpe s ccept wll be te bss te rst clss slut tecques t ee.
2 7. Euler s Metd e rst dervtve gves drect estte te slpe: Were d s evluted t te curret pt. s s Euler s r te Euler- Cuc r pt-slpe etd. w surces errr te plce we slvg rdr deretl equts truct d rud- errr. ruct errr results r te pprt. s tur s tw prts lcl truct errr r ec step d prpgted truct errr tt ccuultes d s crred trug te clcult. e su te tw s te glbl truct errr. e lcl truct errr Euler s etd s O. s c be decresed b tg sller steps d te clcult wll be errr-ree te uct s ler. us Euler s etd s rst-rder tecque d te glbl errr s O. 7.. Iprveets t Euler s Metd e udetl prble wt Euler s etd s tt te dervtve t te begg te tervl s used t estte te slpe crss te etre tervl. Iprveets t Euler s etd cus btg re ccurte represett te verge slpe crss te tervl. Hue s Metd Use te slpe t te begg te tervl t d tl estte r t te ed te tervl te predctr Use ts result t clculte te slpe t te ed te tervl ' Averge tese tw esttes t eld better estte te verge slpe crss te tervl te crrectr
3 [ ] e crrectr c be terted ltug t des t ve t be t bt eve re reed estte r. Wt sll step szes ts tert c qucl cverge d eld better results. I te dervtve s uct l te depedet vrble dd te te predctr d crrectr steps c be cbed t eld s s equvlet t te trpezdl rule uercl tegrt. e lcl errr s terere O wt glbl errr O secd-rder etd. Mdpt r Iprved Plg Metd Euler s etd c be used t predct te vlue t te dpt te tervl e slpe t te dpt te tervl s te Ad c be used t represet te verge slpe crss te tervl Nte tt ts etd des t vlve predctrcrrectr d tus ct be terted t ceve better ccurc. s etd s equvlet t te pe -pt Newt-Ctes tegrt rul d d lcl errr O wt glbl errr O.
4 7. Ruge-Kutt Metds ese re clss tecques tt ceve g ccurc wtut te use g rder dervtves. All re te geerl r φ Were te creet uct φ s te r q q q p q q p q p K M K φ e cecets p d q re detered b settg equl t ters lr eps. e rst-rder Ruge-Kutt etd reduces edtel t Euler s etd. 7.. Secd-Order Ruge-Kutt Metds Usg te lr eps d te secd-rder Ruge-Kutt etds ve te llwg r: q p S tt p ½ d q ½. All te secd-rder Ruge-Kutt etds ve lcl errr O d glbl errr O. Heu s Metd wt sgle crrectr ½
5 5 Mdpt Metd Rlst s Metd u bud te truct errr 7.. rd-order Ruge-Kutt Metd e trd-rder Ruge-Kutt etd s lcl errr O d glbl errr O : 6 I te slpe s uct l te depedet vrble ts etd reduces t te Sp s Rule. 7.. Furt-Order Ruge-Kutt Metd e urt-rder Ruge-Kutt s lcl errr O 5 d glbl errr O d s b r te st ppulr te R-K etds:
6 6 6 s etd ls reduces t te Sps s Rule te slpe s uct l te depedet vrble. Hger-rder Ruge-Kutt etds est but due t prgrg cplet d ccurc requreets te re seld used. 7. Sstes Sulteus Ordr Deretl Equts d d d d d d K M K K Nte tt we wll eed tl cdts t strt te clcults. ese sstes equts re strgt rwrd t pleet cre ust be te t crrectl clculte d ppl te slpes t te vrbles. 7. Adptve Ruge-Kutt Metds Se rdr deretl equts ve ucts tt cge grdull ver prt te d llwg te use lrge step szes d regs rpd cge wc requres te use sller re precse steps. Algrts tt djust te step sze s ecessr re clled dptve d requre estte te lcl errr t ec step t ppl dptve step-sze ctrl. w prr pprces est. e rst esttes te errr s te derece tw predcts usg deret step szes. e secd esttes te errr s te dereces usg tw deret rders te Ruge-Kutt etd.
7 7.. Adptve Ruge-Kutt r Step-Hlvg Metd s etd vlves tg ec te step twce ce sgle ull step d g s tw l steps. e derece betwee te tw esttes s tsel esure te lcl truct errr: Nt l c ts clcult be used r dptve ctrl te step sze t c ls be used t crrect te re precse estte: 5 Ad prvde prved lcl estte O Ruge-Kutt Felberg s pprc uses te derece betwee te t-rder R-K d te urt-rder R-K etds t estte te errr t ec te step. ese tw etds re used becuse te ppe t sre se clcults d l s ttl uct evluts re requred t ec te step t cplete bt etds. See te tet r detls. 7.. Step Sze Ctrl Oce te lcl errr s bee estted e c decde t crese te step sze te lcl errr s sll r decrese te step sze te lcl errr eceeds speced tlerce. Oe strteg s gve s ew ld ew preset α Were α. te step sze s cresed preset ld d α.5 te step sze s decresed preset > ld. ew s usull relted t reltve errr level r ew ε scle were ε s verll tlerce level d scle t gve rctl reltve errrs. Ater relble w t d ts s t set 7
8 scle d d A r spler d re useul tecque spl creses te step sze s reltve errr s lw d decreses te step sze c we te ew c we te errr s lrge. Useul ew ld vlues c rge geerll betwee step sze ctrl d. It suld be ted tt we te errr t gve step s detered t be t lrge tt step suld be reclculted wt sller step utl te errr lls belw te speced tlerce d te slut c prceed. 7.5 Stess St sstes re tse tt superpse rpdl cgg cpets d slwl vrg cpets t ever te step. Mst te te rpdl cgg cpets re trsets tt de ut qucl. e dcult s tt te rpdl vrg prts requre sll te steps d sce te re lws preset dptve step sze ctrl des t wr. Isted te eplct etds eed tus r plct etds setes wr well. Csder te bcwrd r plct Euler s etd wc evlutes te dervtve t uture te: d dt I te geeus prt te slpe s ddt - te ld r Wc s ucdtll stble s. s etd s st -rder ccurte. Hwever plct rults grw cplet s te rder creses d eve re r ler ODEs. Ger devsed set plct rults tt ve lrge stblt lts bsed bcwrd derece rults. ese re te st wdel used etds t slve st sstes. 7.6 Multstep Metds 8
9 9 Multstep etds d t use rt t sgle pt t predct te depedet vrble t uture pt. Isted te use severl prevus pts t detere lel trjectr r te et pt e N-Sel-Strtg Heu Metd Recll tt Heu s etd uses Euler s etd s predctr rwrd derece Ad te trpezdl rule s crrectr e predctr s O wle te crrectr s O tus te predctr s te we l te prcess especll sce te tertve crrectr s depedet te ccurc ts tl predct. d predctr tt s O use te prevus pt - cetrl derece s s w O but uses step sze tt s twce s lrge. Nte tt - s t vlble t te begg te clcult s ts etd s t sel-strtg. I geerl j j j : crrectr : predctr K e crrectr s terted tes usull t eug t ctull cverge but l t prve te predct. pcll. e truct errr per step c be estted s 5 c E Wc c be used t develp ders r te predctr d crrectr:
10 5 : predctr der 5 : crrectr der Nte tt te predctr der uses vlues r te prevus step. e use tese ders s ptl but c speed up cvergece. Step Sze Ctrl Cstt step sze Es t pleet but cce ust be de sll eug r cvergece wt terts. Re-ru te prble lvg te step sze ec te utl cvergece s reced. Vrble step sze Mtr te uber terts r cvergece crrectr d djust s tt. Altertvel duble r lve te step sze tg te uber terts t cverge te crrectr Hger-Order Multstep Metds Hger-rder ultstep etds re bsed pe d clsed Newt-Ctes r Ads tegrt ruls. Ads ruls use severl pts t estte te tegrl l te lst seget te tervl ledg t slgtl re ccurc. Mle s Metd s etd uses -pt Newt-Ctes pe rul s predctr Ad -pt Newt-Ctes clsed rul s crrectr j j ere re wever stblt prbles wt ts crrectr. Usg re stble crrectr leds t Hg s etd: [ ] 9 8 j j
11 Wt crrectrs 9 E p ; Ec Furt-Order Ads Metd s etd uses t -rder Ads-Bsrd pe rul s predctr j d t -rder Ads-Mult clsed rul s crrectr wt ders j j 5 7 ; 9 7 E p Ec 7.7 Budr Vlue Prbles Up t w we ve delt wt tl-vlue prbles were tl cdts speced t e pt re sucet t detere te cstts tegrt d cplete te slut. Prbles tt spec cdts t etree pts r budres te sste re clled budr-vlue prbles. As eple csder te et blce lg t rd d d A L Were s te et trser cecet tt descrbes te rte et dsspt t te surrudg r d s te teperture te surrudg r. slve ts prble tw budr cdts ust be speced r eple te tepertures t te eds te rd L
12 Fr - rd wt C C C d. te slut s 7.5e. 5.5e. C 7.7. e Stg Metd s etd trets budr-vlue prble s equvlet tl-vlue prble. Spec ll te budr cdts t C d e tl guess t dd z te slve te prble t d te teperture t L. I te slut des t tc te budr cdt t L djust te budr cdt t d re-slve te prble. I te prble s ler s ts eple btg tw sluts d terpltg betwee te wll suppl te ecessr budr cdts t. I te prble s -ler r eple te llwg better pprt r te et trser r te br s used d d te stg etd c be cst s geerl rt-dg prble t detere te pprprte strtg vlues rder t rrve t te ter budr vlues. e stg etd s strgtrwrd but becuse te eed t repetedl slve te prble t s t prtculrl ecet Fte Derece Metds Fte derece etds dscretze te d te slut d trsr te ler deretl equt t set sulteus ler equts. Fr te et trser eple bve dvde te rd t equl-legt segets 5 te eple ere. e dvded-derece pprt s d d e deretl equt te beces A 5 L
13 r Applg ts equt t te terr des te rd ssug tt te teperture s w t te eds d L elds 5 e st c budr cdt s were te depedet vrble s speced te budr s te eple bve. ppl tese ed r Drclet budr cdts replce te vrbles wt ter w vlues d elte te equt t tt de. Oe c ls spec te dervtve te depedet vrble t te budr des. Fr eple te et lu dd c be speced t r L bve. ese grdet r Neu budr cdts re ls pprted wt te dereces. Fr eple let te budr cdt dd be speced t bve. g cetrl derece pprt t gves d d Substtutg ts pprt t te te derece equt r de eltes te cttus de t - d gves d d e sste equts r te dl tepertures te beces
14 d d 5 ese sstes equts c be slved r te uw tepertures b te etds dscussed prevusl Egevlue Prbles Egevlue r crcterstc-vlue prbles re clss budr vlue prbles c vbrts elstct d res tt del wt sclltg sstes. ese prbles ve te geerl r [ A λ I ]{ X } Were λ re te egevlues d X re te sscted egevectrs. Pll Metd Develp te set equts [ I ]{ X } A λ. Epd te detert A λi wc wll be pll wse rts re λ. Slve r te rts wt eter Müller s r Brstw s etd deltg rder t d ll te egevluesegevectrs tere wll be e r ec rw A. Pwer Metd Wrte te sste s {... } AX λx. Fr tl guess ssue tt X substtute d slve r ew set X. Nrlze wt respect t te lrgest vlue X. Iterte utl cvergece. Up cvergece te rlzt ctr wll be te lrgest egevlue wt egevectr equl t X. I tr A s setrc t c te be delted usg Htellg s etd A A λ X X were A s te rgl tr d λ X re te lrgest egevlueegevectr pr. Prceed ts er t d te lrgest severl egevlues. s etd ct usull be used t d ll te egevlues due t te ccuult sgct rud- errrs. d te sllest egevlueegevectr prs perr te pwer etd te verse A deltg rder t elte tse lred ud.
15 M re dvced tecques est r dg egevlues. 5
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