PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION) M E T O D E E U L E R M E T O D E R U N G E - K U T T A

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1 PERSAMAAN DIFERENSIAL DIFFERENTIAL EQUATION M E T O D E E U L E R M E T O D E R U N G E - U T T A

2 PERSAMAAN DIFERENSIAL Persamaan palng pentng dalam bdang reaasa palng bsa menjelasan apa ang terjad dalam sstem s. Mengtung jara teradap atu dengan ecepatan tertentu 5 msalna. d dt 5

3

4 Rate equatons

5 PERSAMAAN DIFERENSIAL Solusna secara analt dengan ntegral d 5 5t C dt C adala onstanta ntegras Artna solus analts tersebut terdr dar bana alternat C ana bsa dcar ja mengetau nla dan t. Sengga untu conto d atas ja = saat t= = maa C =

6 LASIFIASI PERSAMAAN DIFERENSIAL Persamaan ang mengandung turunan dar satu atau leb varabel ta bebas teradap satu atau leb varabel bebas. Dbedaan menurut: Tpe ordner/basa atau parsal Orde dtentuan ole turunan tertngg ang ada Lnart lner atau non-lner

7 SOLUSI PERSAMAAN DIFERENSIAL Secara analt mencar solus persamaan derensal adala dengan mencar ungs ntegral na. Conto untu ungs pertumbuan secara esponensal persamaan umum: dp dt P

8 Rate equatons

9 But at ou reall ant to no s te szes o te boes or state varables and o te cange troug tme Tat s ou ant to no: te state equatons Tere are to basc as o ndng te state equatons or te state varables based on our non rate equatons: Analtcal ntegraton Numercal ntegraton

10 Suatu ultur batera tumbu dengan ecepatan ang proporsonal dengan jumla batera ang ada pada setap atu. Detau baa jumla bater bertamba menjad dua al lpat setap 5 jam. Ja ultur tersebut berjumla satu unt pada saat t = berapa ra-ra jumla bater setela satu jam?

11 SOLUSI PERSAMAAN DIFERENSIAL Jumla bater menjad dua al lpat setap 5 jam maa = ln /5 Ja P = unt maa setela satu jam P dp dt P t dp dt P P t P ln C t t P P t P e P e t ln 5.487

12 Te Analtcal Soluton o te Rate Equaton s te State Equaton Rate equaton dsolve n Maple State equaton

13 THERE ARE VERY FEW MODELS IN ECOLOGY THAT CAN BE SOLVED ANALYTICALLY.

14 SOLUSI NUMERI Numercal ntegraton Eulers Runge-utta

15 Numercal ntegraton maes use o ts relatonsp: tt t d dt t Wc ou ve seen beore Relatonsp beteen contnuous and dscrete tme models *You used ts relatonsp n Lab to program te logstc rate equaton n Vsual Basc: N t N t Nt rnt t ere t

16 Fundamental Approac o Numercal Integraton tt t d dt t t+t unnon = t unnon t+t estmated t non t speced d dt non t

17 Nt/ N tt N t rn t N t t ere t dn dt N t / t tme lambda =.7 tme step = tme ears Calculate dn/dt* at N t Add t to N t to estmate N t+ t N t+ t becomes te ne N t Calculte dn/dt * at ne N t Use dn/dt to estmate net N t+ t Repeat tese steps to estmate te state uncton over our desred tme lengt ere ears Euler s Metod: t+ t t + d / dt t

18 EXAMPLE OF NUMERICAL INTEGRATION d dt 6. 7 pont to estmate Analtcal soluton to d/dt Y = t =.5

19 Euler s Metod: t+ t t + d / dt t d dt 6. 7 m = d/dt at t m = 6*-.7* = m *t est = t + analtcal t+ t estmated t+ t t = t =.5

20 RUNGE-UTTA METHODS

21 MOTIVATION We see accurate metods to solve ODEs tat do not requre calculatng g order dervatves. Te approac s to use a ormula nvolvng unnon coecents ten determne tese coecents to matc as man terms o te Talor seres epanson.

22 SECOND ORDER RUNGE-UTTA METHOD possble. as accurate as s suc tat Problem : Fnd

23 TAYLOR SERIES IN ONE VARIABLE Te n t order Talor Seres epanson o n! n n! n Appromaton Error ere s beteen and

24 DERIVATION OF ND ORDER RUNGE-UTTA METHODS OF 5 4 ' rttenas : c s Used tosolve ODE: Epanson Seres Order Talor Second O O d d d d d d

25 DERIVATION OF ND ORDER RUNGE-UTTA METHODS OF 5 5 : ' rule derentaton - can obtaned b s ' ere O Substtutng d d

26 TAYLOR SERIES IN TWO VARIABLES 6 and beteen jonng on telne s!!...! error appromaton n n n

27 DERIVATION OF ND ORDER RUNGE-UTTA METHODS OF 5 7 Substtutng : suc tat Problem : Fnd

28 DERIVATION OF ND ORDER RUNGE-UTTA METHODS 4 OF :... Substtutng

29 DERIVATION OF ND ORDER RUNGE-UTTA METHODS 5 OF 5 9 One possblesoluton: nnte solutons t 4 unnons equatons and e obtan te ollong tree equatons : M atcng terms... : or toepansons We derved O

30 ND ORDER RUNGE-UTTA METHODS and : suc tat Coose

31 ALTERNATIVE FORM Order Runge utta Second Alternatve Form

32 CHOOSING W AND W Corrector t a Sngle ' Ts s : metod becomes utta Order Runge - Second ten coosng For eample s Metod Heun

33 CHOOSING W AND W te M dpontm etod Ts s : metod becomes utta Order Runge - Second ten Coosng

34 ND ORDER RUNGE-UTTA METHODS ALTERNATIVE FORMULAS 4 select Order Runge utta Formulas Second : number an nonzero Pc

35 SECOND ORDER RUNGE-UTTA METHOD EXAMPLE CISE_Topc /.8 4 / STEP:. 4 usng R. te ollong sstemto nd Solve t t t t t t

36 SECOND ORDER RUNGE-UTTA METHOD EXAMPLE STEP t t t t

37 t t t 4 Soluton or t [] Usng R 7

38 ND ORDER RUNGE-UTTA R Tpcalvalue o no as R Equvalent to Heun' s metod t a sngle corrector Local error s O and global error s O 8

39 HIGHER-ORDER RUNGE-UTTA Hger order Runge-utta metods are avalable. Derved smlar to second-order Runge-utta. Hger order metods are more accurate but requre more calculatons. 9

40 RD ORDER RUNGE-UTTA 4 R error s and Global error s Local 4 6 R no as 4 O O

41 4 TH ORDER RUNGE-UTTA 4 R4 error s global and error s Local O O

42 HIGHER-ORDER RUNGE-UTTA

43 EXAMPLE 4 TH -ORDER RUNGE-UTTA METHOD R4 d d.5. Use R 4to compute. and.4 4

44 EXAMPLE: R4 Problem : d d Use R 4 to nd

45 4 TH ORDER RUNGE-UTTA 45 R4 error s global and error s Local O O

46 EXAMPLE: R Problem : nd to R Use d d See R4 Formula Step

47 EXAMPLE: R Problem : nd to R Use d d Step

48 EXAMPLE: R4 Problem : d d Use R 4 to nd.5..4 Summar o te soluton

49 SUMMARY Runge utta metods generate an accurate soluton tout te need to calculate g order dervatves. Second order R ave local truncaton error o order O and global truncaton error o order O. Hger order R ave better local and global truncaton errors. N uncton evaluatons are needed n te N t order R metod. 49