ORDINARY DIFFERENTIAL EQUATIONS EULER S METHOD

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1 Numercal Analss or Engneers German Jordanan Unverst ORDINARY DIFFERENTIAL EQUATIONS We wll eplore several metods o solvng rst order ordnar derental equatons (ODEs and we wll sow ow tese metods can be appled to sstems o rst order derental equatons as well as ger order ODEs. Pscal sstems are usuall represented b derental equatons. For eample te equaton o moton o a mass-sprng-damper sstem s d m c dt d dt Our nterest s n te beavor o as a uncton o tme (t. In te ollowng metods we wll concentrate on solvng rst order ODEs n te orm EULER S METHOD d d ( d Te slope s used to predct a new value startng rom an ntal pont ( d Usng nte orward dvded derence te slope at pont ( s Ten d d ( (. stes.google.com/ste/zadmasoud/numercal

2 Numercal Analss or Engneers German Jordanan Unverst Eample For te ollowng sstem determne te value o rom.5 and te ntal condton ( Te eact soluton s : d d to 4 usng a step sze o Usng Euler s metod te ntal condton s used to determne te ntal slope or computng te rst predcted value. Step one and Te slope at s stes.google.com/ste/zadmasoud/numercal

3 Numercal Analss or Engneers German Jordanan Unverst Te rst predcted value s Te true value at. 5 s Te true error s d d ( ( ( ( ( 8.5 ( (.5 4 4(.5 (.5 8.5(.5.9 E t true appromat e and te percent relatve true error s 6.9% Step two.5 and 5. 5 Te predcted value s 5.5 Te true value at s ( ( t (.5 Te percent relatve true error s 95.8% ( ( ( 4 4( ( 8.5( t Computatons are repeated untl 4. Results are sown n te ollowng table. stes.google.com/ste/zadmasoud/numercal

4 Numercal Analss or Engneers German Jordanan Unverst true (% t Te smaller te step sze s te smaller te error. Usng step sze o.5 n te prevous eample te true percent relatve true error at 4 s.%. Eample For te ollowng sstem determne te value o rom. and te ntal condton ( Te eact soluton s Step one : d d e ( t to usng a step sze o stes.google.com/ste/zadmasoud/numercal

5 Numercal Analss or Engneers German Jordanan Unverst and Te true value at. s ( (.... e...4 Te percent relatve true error s.8% Step two. and. Te predcted value s Te true value at. 4 s t. ( (..48 e Te percent relatve true error s 6.566%. Results are sown n te ollowng table. t stes.google.com/ste/zadmasoud/numercal 4

6 Numercal Analss or Engneers German Jordanan Unverst true (% t ERROR ANALYSIS FOR EULER S METHOD In te numercal soluton o ODEs two tpes o error are encountered; round-o errors and truncaton errors. To determne te truncaton error or Euler s metod we wll go bac to Talor seres epanson rom wc te metod was derved. were R ( n! n ( n!! ( n n n! n ( and Euler s algortm s bascall te rst order Talor epanson were ( R n stes.google.com/ste/zadmasoud/numercal 5

7 Numercal Analss or Engneers German Jordanan Unverst Tereore te truncaton error s Ea!! ( n n O( n! For small step sze te truncaton error can be appromated as E a (! O( Te truncaton error s nown as te local error wc s te error ntroduced n eac ntegraton step. Te true error calculated n te earler eamples s nown as te global error wc can be sown to be O (. Eample For te ollowng sstem determne te value o rom. and te ntal condton ( and Te truncaton error s were (. Tereore : d d E ( a ( ( (..4.. ( n to usng a step sze o stes.google.com/ste/zadmasoud/numercal 6

8 Numercal Analss or Engneers German Jordanan Unverst Usng more terms n te truncaton error E a E!! ( Te truncaton error o te rst step s! ( (. (4 4! 4 (! 4! 4 (!.4 ( 4 4! (. (. E (.467!! wc s n te same order o magntude as te one term truncaton error computed earler. Eample For te ollowng sstem determne te value o rom. and te ntal condton ( and : d d ( ( ( (..6.. to usng a step sze o stes.google.com/ste/zadmasoud/numercal 7

9 Numercal Analss or Engneers German Jordanan Unverst Ten E d ( ( d d d ( ( (. Complete results are sown n te ollowng table. Eample Ea ( For te ollowng sstem determne te value o rom. and te ntal condton ( and : d d ( ( (..8 (... to usng a step sze o stes.google.com/ste/zadmasoud/numercal 8

10 Numercal Analss or Engneers German Jordanan Unverst. and. 8.8 (..8 ( ( Complete results are sown n te ollowng table HEUN S METHOD (PREDICTOR-CORRECTOR METHOD p In ts metod an ntal appromaton o te value o te uncton at s determned usng Euler s metod. Ts ntal appromaton s used to compute te slope o te uncton at. Ten te average o te slopes at and s used agan wt Euler s metod to determne te nal value o te uncton at. Te average slope s stes.google.com/ste/zadmasoud/numercal 9

11 Numercal Analss or Engneers German Jordanan Unverst stes.google.com/ste/zadmasoud/numercal 4 av were ( ( p Te ntal predcted value o te uncton p usng te Predctor (Euler p ( Ten te value at s corrected usng te Corrector (Euler p ( ( TRUNCATION ERROR Usng Talor seres epanson

12 Numercal Analss or Engneers German Jordanan Unverst O( Usng te nd nte dvded derence to appromate te local truncaton error s O( O( It can be sown tat te global truncaton error o te Heun s metod s O ( APPROXIMATE ERROR Snce two values o te uncton are computed at an appromate error can be calculated as a p % Eample For te ollowng sstem determne te value o rom. and te ntal condton ( Te slope o te gven uncton s : d d ( to usng a step sze o stes.google.com/ste/zadmasoud/numercal 4

13 Numercal Analss or Engneers German Jordanan Unverst stes.google.com/ste/zadmasoud/numercal 4 and...8 (. ( ( p Ten.856 (..8 (. ( ( ( p and 6.54% % % p a.856. and (..856 (..856 ( p Ten.7866 ( ( (..856 ( ( p

14 Numercal Analss or Engneers German Jordanan Unverst and a p % Complete results are sown n te ollowng table % 4.76% a (% MIDPOINT METHOD (MODIFIED EULER In ts metod te value o te uncton s predcted at te mdpont o te nterval usng Euler s metod. A new slope or te uncton s determned at te mdpont usng te predcted value o te uncton. Ten ts new slope s used wt Euler s metod to determne te value o te uncton at te end o te nterval. Usng Euler s metod te value o te uncton usng al o te step sze s and ( stes.google.com/ste/zadmasoud/numercal 4

15 Numercal Analss or Engneers German Jordanan Unverst Te slope at te mdpont s ten ( Now usng ts new slope wt Euler s metod te value o te uncton at s Eample ( For te ollowng sstem determne te value o rom. and te ntal condton ( Te slope o te gven uncton s : d d ( to usng a step sze o stes.google.com/ste/zadmasoud/numercal 44

16 Numercal Analss or Engneers German Jordanan Unverst and Ten. and.858 Ten ( (...9 ( (..9 (.858 ( ( (.858 (..844 Complete results are sown n te ollowng table ( stes.google.com/ste/zadmasoud/numercal 45

17 Numercal Analss or Engneers German Jordanan Unverst FOURTH ORDER RUNGE-KUTTA METHOD Runge-Kutta (RK metod s te most wdel used metods. Te general orm o te (RK metod s were and ( a ( ( ( ( a p p Te 4 t order RK metod uses terms troug 4. Te most commonl used set o value s a a a a a n n stes.google.com/ste/zadmasoud/numercal 46

18 Numercal Analss or Engneers German Jordanan Unverst were ( 6 4 ( ( ( ( It can be sown tat te local error o te RK4 s O ( 5 and te global error s ( 4 Eample For te ollowng sstem determne te value o rom. and te ntal condton ( Te slope o te gven uncton s and 4 ( ( ( ( : d d. (. (. ( (.(. 4 to. (..4. ( ( ( O. usng a step sze o stes.google.com/ste/zadmasoud/numercal 47

19 Numercal Analss or Engneers German Jordanan Unverst Ten. and.85 Ten 4 ( 4 6 ((. (.4 (.56 ( ( ( ( (.(..85. (.. (.. (..49. ( ( ( ( ( 4 6 ((.49 (.676 (.76 ( Complete results are sown n te ollowng table stes.google.com/ste/zadmasoud/numercal 48

20 Numercal Analss or Engneers German Jordanan Unverst Te eact soluton up to sgncant gures s (.889 stes.google.com/ste/zadmasoud/numercal 49

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