ORDINARY DIFFERENTIAL EQUATIONS

Transcription

1 6 ORDINARY DIFFERENTIAL EQUATIONS Introducton Runge-Kutt Metods Mult-step Metods Sstem o Equtons Boundr Vlue Problems Crcterstc Vlue Problems

2 Cpter 6 Ordnr Derentl Equtons / 6. Introducton In mn engneerng pplctons tere re lws o pscs wc cn be mtemtcll wrtten n orm o derentl (or rte) equtons e.g. dv dt c v (6.) m Eq. 6. s nown s ordnr derentl equton (ODE) snce tere s onl one ndependent vrble.e. t oterwse t s prtl derentl equton (PDE). Eq. 6. s rst order ODE weres n emple o second order ODE s d d m c (6.) dt dt Solvng Eqs. 6.- needs set o condtons wc cn be dvded nto:. Intl vlue problems onl one condton s requred. Boundr vlue problems onl more tn one condtons re requred.

3 Cpter 6 Ordnr Derentl Equtons / 6. Runge-Kutt Metods Te Runge-Kutt metod s one-step metod wc requres one ntl condton: d d ( ) ; ( ) A generl numercl ormul berng to solve te bove equton s (New vlue) (old vlue) (Grdent) (Step sze) were s te step sze. (6.) φ I te estmton o grdent ( ) φ s ten rom te rst dervtve t te tecnque s nown s te Euler (or grdent pont) metod nd ( ) (6.) In solvng n ODE tere re two tpes o errors:. Globl error reltve to ect soluton. Locl error reltve to te prevous numercl soluton. To clculte te locl error consder Tlor seres epnson:! n! n n L R n n ( ξ ) ( n )! were R n s te resdul term. I ( ) n : R n ( ) ( ) ( ) L! n Hence te locl truncted error s: ( ) n ( ) n! n O n ( ) E t L O (6.5)!

4 Cpter 6 Ordnr Derentl Equtons / Emple 6. Use te Euler metod to obtn te vlues o numercll or te ollowng derentl equton: d d 8.5 Gven tt te ntl condton s () perorm te clculton rom to wt step sze o.5. For comprson use te nltcl soluton: Soluton Te ODE cn be wrtten s: d d For te rst step (.5): ( ) Vlue usng te Euler metod: Ect vlue: True error: (.5).5(.5) (.5) (.5) 8.5(.5).875. E ε 6.% t For te second step (.5): t (.) (.5) (.5 5.5) 5.5 [ (.5) (.5) (.5) 8.5](.5) 5.875

5 Cpter 6 Ordnr Derentl Equtons / 5 Euler ect Globl error Locl error Anltcl soluton FIGURE 6. Grp or E. 6. For E. 6. te truncted locl error cn be clculted t s ollowed: E t E t ( ) ( ) ( )!! ( ) 6 ( ) ( ) 6 (.5) (.5) (.5) 6.5.!

6 Cpter 6 Ordnr Derentl Equtons / 6 Te Euler metod cn be mproved usng smller step sze. 8.5 Anltcl soluton.5 FIGURE 6. Eect o usng smller step sze or E. 6. To urter mprove te tecnque te Euler metod cn be used or predcton wc s to be corrected v terton mecnsm. In te predctor-corrector metod te predctor equton s (6.6) Use to obtn te grdent t nd ten te te verge: φ φ φ φ Hence te corrector equton s φ φ φ or (6.7)

7 Cpter 6 Ordnr Derentl Equtons / 7 B tng te verge o grdents Eq. 6.7 s nown s te Heun metod wc cn be solved tertvel nd generlsed s j ε j ( ) ( ) j j j % Emple 6. Solve te ollowng equton or te rnge o to : d d e. 8.5 wt n ntl condton o () nd step sze o. Perorm clculton or 5 tertons. Soluton At : Predctor: d.8 φ e d Grdent vergng: e.5 5 ( ) ( 5) φ.8 ().5 () φ.78 Corrector: ε 5.% 6.78 In te net terton: ε 6.78% [ ] e () 6.758

8 Cpter 6 Ordnr Derentl Equtons / 8 ect Ater terton Ater 5 tertons Heun ε t % Heun ε t % Te generl equton or te Runge-Kutt metods s Te grdent term ( ) wrtten s ( ) φ (6.8) φ s nown s terton uncton wc cn be φ L n n were re constnts nd re dened s n M ( ) ( p q) ( p q q ) ( p q q L q ) n n n n n Wen n te Euler metod cn be ormed wc represents te rst order Runge-Kutt metod. For n te second order Runge-Kutt metod cn be ormed: ( ) n (6.9) were ( ) ( p q )

9 Cpter 6 Ordnr Derentl Equtons / 9 TABLE 6. Te second order Runge-Kutt metods Metod p q Equton Heun Mdpont Rlston ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Emple 6. Use te Heun mdpont nd Rlston metods to solve E. 6.. Soluton Te Heun metod: ( ) ( ) (.5) (.5) (.5) Hence [ ( 8.5) (.5) ] ε 6.8% Te mdpont metod: Hence t.5.75 ( ) (.5) s (.5) (.5).875 (.5).875(.5) ε.% t

10 Cpter 6 Ordnr Derentl Equtons / Te Rlston metod: Hence ( ) (.75) (.75) (.75).585 [ ( 8.5) (.585) ] ε.8% t sebenr Heun Mdpont Rlston ε t % ε t % ε t % For n te trd order Runge-Kutt metod cn be ormed: were ( ) (6.) 6 ( ) ( ) ( ) For n te ourt order Runge-Kutt metod cn be ormed: ( ) (6.) 6

11 Cpter 6 Ordnr Derentl Equtons / were For n 5 te t order Runge-Kutt metod (lso nown s te Butcer metod) cn be ormed: (6.) were

12 Cpter 6 Ordnr Derentl Equtons / 6. Mult-step Metods In te mult-step metods te clculton s bsed on more tn one ponts nd populr verson s te Adms ormul. () One-step metod (b) Mult-step metod FIGURE 6. Comprson between te one-step metod nd te mult-step metod It cn use te predctor nd corrector pproces wc re termed s te open nd closed ormule. Te open ormul s nown s te Adms-Bsort predctor wc cn be derved rom te Tlor seres t : 6 L L 6 Tng O O L wc cn be summrsed n orm o second order equton: 5 ( ) O( ) (6.) 6

13 Cpter 6 Ordnr Derentl Equtons / Te generl orm o te n-t order Adms-Bsort predctor s: n n ( ) β O (6.) TABLE 6. Coecents or te Adms-Bsort predctor Order β β β β Error ( ξ ) ( ξ ) ( ξ ) ( ) ( ξ ) Te closed ormul s nown s te Adms-Moulton corrector wc cn be derved rom te Tlor seres t : 6 6 L L Tng wc produces second order equton: O ( ) O (6.5) Te generl orm o te n-t order Adms-Moulton corrector s: n n ( ) β O (6.6)

14 Cpter 6 Ordnr Derentl Equtons / TABLE 6. Coecents or te Adms-Moulton corrector Order β β β β Error - - ( ξ ) ( ξ ) ( ) ( ξ ) Emple 6. Use te combnton o te ourt order Adms-Bsort predctor nd te Adms Moulton corrector to solve te ollowng equton t : d d e. 8.5 Gven tt te ntl condton s () nd te step sze s nd te normton t te tree prevous ponts re: : : : As gudnce te ect soluton s: Soluton. Te vlues o dervtves t ll ponts: e e e e ( e e ) e ( ).5 ( ) ( ).5 (.9995 ) ( ).5 (.66 ) Ten use te Adms-Bsort predctor: (.57) [ ] 9 [ ] () () 6.756

15 Cpter 6 Ordnr Derentl Equtons / 5 Comprson wt te ect soluton:. () e.5 () () ().8 e e.5 ε t.% Net use te Adms-Moulton corrector: Hence.8 e.5 e ().5( 6.756) [ ] [ ] () () 6.5 ε t.96%

16 Cpter 6 Ordnr Derentl Equtons / 6 6. Sstem o Equtons Generll sstem o ODEs contnng n equtons (tus requres n ntl condtons) cn be wrtten s: d d d d dn d M n ( L ) ( L ) ( L ) n n n (6.7) Te concept o sstem o ODEs cn lso be used to solve ger order ODE b decomposng nto severl rst order ODEs or emple: d d d d d b d c ( ) vng tree ntl condtons: At d d α β γ d d : cn be converted nto: d d d d d d ( ) b c nd te tree ntl condtons now become: At : α β γ

17 Cpter 6 Ordnr Derentl Equtons / 7 Emple 6.5 Use te Euler metod to solve te ollowng sstem: d d d d.5.. rom to. Use te ntl condtons o () nd () 6 nd step sze o.5. Soluton For te rst step: Te net steps re s ollowed: (.5) [.5 ].5 (.5) 6 [.( 6). ]

18 Cpter 6 Ordnr Derentl Equtons / 8 Emple 6.6 Repet E. 6.5 usng te mdpont metod. Soluton For te rst step: Ten t.5: ( 6). [ ( )(.5) ] (.8)(.5) Te net steps re s ollowed:.8.75 [ ].[ ( )(.5) ]. 75 (.5) (.75)(.5).5 (.5) 6 (.75)(.5)

19 Cpter 6 Ordnr Derentl Equtons / Boundr Vlue Problems An ODE o n order more tn one usull requre more tn one condton suc problem s reerred to s te boundr vlue problem. For emple second order ODE requre two condtons nd cn be wrtten n orm o d d ( ) (6.8) wt te boundr condtons: : L : L For ts problem te nte derence metod cn be used were te second order dervtves cn be dscretsed v centrl derence s: d d Δ ( Δ) Δ Δ Hence Eq. (6.8) becomes ( Δ) ( ) (6.) Te ccurc o ts metod depends on te selected step sze Δ. Emple 6.7 Obtn temperture dstrbuton long slender rod sown n Fg. 6. were ts terml bevour cn be represented b te ollowng equton: d T d ( T ) α T Its boundr condtons re T() C nd T(L) C. Use L m α. m T C nd step sze o m.

20 Cpter 6 Ordnr Derentl Equtons / Ambent temperture C T C T C FIGURE 6. A slender rod vng terml loss troug convecton For comprson te nltcl soluton s Soluton T 7.5e. 5.5e Te dscretsed equton cn be rewrtten s T T T. T.T T Ten te ollowng equtons cn be ormed: or n mtr orm: T T.T.T.T T T T.T T... T T Comprson wt nltcl soluton:.8.8. ( T ) T.8.8 T T.8. T.8 T.58 T T T ect ε t (%)

21 Cpter 6 Ordnr Derentl Equtons / 6.6 Crcterstc Vlue Problems Tere re problems n elstct nd vbrton wc re ctegorsed s crcterstc vlue problems wen te second order ODEs re solved n ter omogeneous orm. For emple consder Eq. 6. or sstem o sprng wtout dmpng vng one sngle degree o reedom: or n noter orm: d m (6.) dt d dt ω (6.) were ω m s te nturl requenc o te sstem. Te generl soluton o Eq. 6. s Asn ωt B cosωt B usng te boundr condtons n Eq. 6.: Hence ω ± mπ m K Asn mπt For sstem vng n degree o reedom te mss nd stness propertes cn be represented n orm o mtrces M nd K respectvel: d M K dt (6.) Usng te smlr generl soluton s bove Eq. 6. cn be converted to [ K ] M {} {} ω (6.) Pre-multplcton o Eq. 6. wt M produces generl egen equton: [ K ω M] {} [ M K ω M M] {} {} M

22 Cpter 6 Ordnr Derentl Equtons / [ A I] { } { } λ (6.5) were A M K s squre mtr nd λ ω s te egen vlue or A. For non-zero soluton o : ( A I) A λi det λ (6.6) A λ (6.7) Eq. 6.6 s nown s te crcterstc equton nd ts soluton produces n egen vlues (rel or comple) nd te correspondng solutons wc ollows Eq. 6.7 s clled egen vector. To obtn egen vlues Eq. 6.6 cn be rewrtten s n λ ( λ) λ L n λ n M M M n L L O L nn n n n M λ (6.8) wc cn be epnded nto n n-t order polnomls o λ. To obtn te mmum rel egen vlue te power metod cn be used were Eq. 6.7 cn be moded s ollowed λ A (6.9) Durng te clculton te vector s normlsed suc tt ts mmum component s lws unt. Emple 6.8 Obtn te mmum egen vlues nd te correspondng egen vector usng te power metod: Perorm terton untl te soluton converges t decml plces. Use n ntl vector o v ( ) T.

23 Cpter 6 Ordnr Derentl Equtons / Penelesn For te rst terton:.5 For te second terton: For te trd terton: For te ourt terton: And t converges ter 5 tertons: Hence te mmum egen vlue s 5.77 nd te correspondng egen vector s (.7.) T. I tere s need to determne te mnmum rel egen vlue te power metod cn be moded to orm te nverse power metod rom Eq. 6.7: A A A λ A λ (6.)

24 Cpter 6 Ordnr Derentl Equtons / Eercses. Solve te ollowng sstem o ordnr derentl equtons: d d d d rom to usng te Heun metod wt te step sze o.. Plot te grps o nd gnst nd te grp o gnst.. Te ollowng equton cn be used to model te delecton (z) o ed slender column vng wnd lodng: d dz EI ( L z) were E s te modulus o elstct (. 8 N/m ) I s te re moment o nert (.5 m ) L s te column egt ( m) nd s te dstrbuton uncton o wnd orce wc vres wt elevton nd s gven b z 5 z z ( z) e [N m] Perorm te clculton usng te trd order Runge-Kutt metod to obtn te delecton t ll poston long te column. Use te step sze o 5 m.. Obtn te lrgest nd smllest egen vlues or te ollowng mtr s well s te correspondng egen vectors: For ec cse use te ntl egen vector o ( ) T v.