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
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
Rate equatons
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 =
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
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
Rate equatons
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
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?
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
Te Analtcal Soluton o te Rate Equaton s te State Equaton Rate equaton dsolve n Maple State equaton
THERE ARE VERY FEW MODELS IN ECOLOGY THAT CAN BE SOLVED ANALYTICALLY.
SOLUSI NUMERI Numercal ntegraton Eulers Runge-utta
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
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
Nt/ N tt N t rn t N t t ere t dn dt N t / t tme lambda =.7 tme step =.45.4.5..5..5..5 4 5 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
EXAMPLE OF NUMERICAL INTEGRATION d dt 6. 7 pont to estmate Analtcal soluton to d/dt Y = t =.5
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
RUNGE-UTTA METHODS
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.
SECOND ORDER RUNGE-UTTA METHOD possble. as accurate as s suc tat Problem : Fnd
TAYLOR SERIES IN ONE VARIABLE Te n t order Talor Seres epanson o n! n n! n Appromaton Error ere s beteen and
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
DERIVATION OF ND ORDER RUNGE-UTTA METHODS OF 5 5 : ' rule derentaton - can obtaned b s ' ere O Substtutng d d
TAYLOR SERIES IN TWO VARIABLES 6 and beteen jonng on telne s!!...! error appromaton n n n
DERIVATION OF ND ORDER RUNGE-UTTA METHODS OF 5 7 Substtutng : suc tat Problem : Fnd
DERIVATION OF ND ORDER RUNGE-UTTA METHODS 4 OF 5 8......... :... Substtutng
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
ND ORDER RUNGE-UTTA METHODS and : suc tat Coose
ALTERNATIVE FORM Order Runge utta Second Alternatve Form
CHOOSING W AND W Corrector t a Sngle ' Ts s : metod becomes utta Order Runge - Second ten coosng For eample s Metod Heun
CHOOSING W AND W te M dpontm etod Ts s : metod becomes utta Order Runge - Second ten Coosng
ND ORDER RUNGE-UTTA METHODS ALTERNATIVE FORMULAS 4 select Order Runge utta Formulas Second : number an nonzero Pc
SECOND ORDER RUNGE-UTTA METHOD EXAMPLE CISE_Topc8 5.869.66 /.8 4 /..66..8..8. 4 STEP:. 4 usng R. te ollong sstemto nd Solve t t t t t t
SECOND ORDER RUNGE-UTTA METHOD EXAMPLE 6.666.546.668.869....546..668..668..869. STEP t t t t
t t t 4 Soluton or t [] Usng R 7
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
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
RD ORDER RUNGE-UTTA 4 R error s and Global error s Local 4 6 R no as 4 O O
4 TH ORDER RUNGE-UTTA 4 R4 error s global and error s Local 6 4 5 4 4 O O
HIGHER-ORDER RUNGE-UTTA 6 5 4 5 4 6 4 5 4 7 7 9 7 8 7 7 7 7 6 9 6 4 8 8 4 4 4 4
EXAMPLE 4 TH -ORDER RUNGE-UTTA METHOD R4 d d.5. Use R 4to compute. and.4 4
EXAMPLE: R4 Problem : d d Use R 4 to nd.5..4 44
4 TH ORDER RUNGE-UTTA 45 R4 error s global and error s Local 6 4 5 4 4 O O
EXAMPLE: R4 89. 6.798..6545.654..64.64..5.5 4 4.4. 4.5 Problem : nd to R Use d d.5. 46 See R4 Formula Step
EXAMPLE: R4 4. 6..555.9.98.789 4 4.4. 4.5 Problem : nd to R Use d d.89.. 47 Step
EXAMPLE: R4 Problem : d d Use R 4 to nd.5..4 Summar o te soluton..5..89.4.4 48
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