Numerical methods for ordinary differential equations

Transcription

1 A Maser Scieiic copuig Nuerical ehos or oriar iereial equaios

2 Oriar iereial equaios Le be a ocio o oe variable havig successive erivaives. A h orer oriar iereial equaio ODE is a equaio o he or: A s orer ODE is hereore give b: ODE are iere ro parial iereial equaios which ivolve a ucio epeig o ore ha oe variable Fx x x 3 a is parial erivaives E/x E/x...

3 3 Sse o oriar iereial equaios Le be ucios o oe variable havig successive erivaives. A sse o couple oriar iereial equaios o orer is o he or: I ca be wrie :

4 4 Sse o oriar iereial equaios Sipliicaio I we have : The wriig: The sse o ODE o orer ca alwas be wrie as a s ODE : w w w w w w w w w w w w

5 5 Jacobia arix o a s orer ODE Le s cosier he s orer ODE: B eiiio he Jacobia arix o his sse o iereial equaios is he arix give b: J

6 Two ai pes o ODE We have : Iiial value proble IVP : All ucios are kow a = 0 a we look or a soluio a > 0 Mah heore: I is regular C a Lipschizia here exiss a uique soluio. A uerical copuaio o his soluio is usuall eas excep i he Jacobia arix is close o sigular; i his case we sa ha he ODE is si Bouar value proble BVP: Soe ucios soluio or [ ]. are kow a = ohers a = a we look or a No ah heore : Depeig o he ODE here ca be oe a or o soluio. Nuerical copuaio is uch ore iicul ha or a iiial value proble. 6

7 Nuerical resoluio o a ODE Iiial value proble Euler eho : = a 0 = 0 For exaple: si a 0 0 Copuaio o 0 0 Copuaio o he age a 0 0 Esiaio o he soluio a = 0 + D Copuaio o Copuaio o he age a D Esiaio o he soluio a = + D 0 D D 3 Ec This echique gives a approxiaio o he aheaical soluio saple wih a sep D. 7

8 Malab ools or he resoluio o a iiial value proble ODE. No si sses: oe45: oe3: The bes solver o use as a irs r or os probles. Ca be ore eicie ha oe45 or low oleraces a i he presece o il siess. oe3: Ca be ore eicie ha oe45 a srige oleraces. Si sses: oe5s: Shoul be use i a proble is suspece o be si or i oe45 aile or is ver ieicie. oe3s: More eicie ha oe5s a low oleraces. Ca solve soe kis o si probles or which oe5s is o eecive. 8

9 Basic sax or oexx' solvers To solve: [v] = oexx[0 ]0; : reerece o he ucio ucio s=aoc wih scalar colu vecor a s colu vecor. 0: scalar uber correspoig o he iiial ie. : scalar correspoig o he ial ie. 0: colu vecor wih iiial coiios 0 v: ie vecor where he soluio o he ODE has bee copue. : arix o legh: leghv x legh0 coaiig he saplig o he copue soluio plov:k plos or he copoe k o he soluio o [ 0 ] 9

10 Exaple : / / / or [0 0] wih 0 = a 0 = ucio s=ex % Ex reerece o ucio i exaple % : scalar uber % : colu vecor o legh [;] % s: colu vecor o legh [/; /] s=[;-]; % oe: i his exaple oes o explicil appears. =0; =0; 0=[;-]; [v]=oe45@ex[ ]0; plov xlabel'' label'' lege'_''_' 0

11 Exaple : /..e /. /..e. or [0 0] wih 0 = a 0 = ucio s=exalphabea % Ex reerece o ucio i exaple % : scalar uber % : colu vecor o legh [;] % alpha bea: scalar ubers paraeers o he proble. % s: colu vecor o legh [/; /] s=[alpha**exp-;-bea*]; =0; =0; 0=[;-]; alpha=; bea=3; Exab=@ Exalphabea; % aoous paraeric ucio hale [v]=oe45exab[ ]0; plov xlabel'' label'' lege'_''_' ile'\alpha= \bea=3'

12 Malab ools or he resoluio o bouar value proble ODE : bvp4c a bvp5c sol = bvp4or 5cgsoliiopios Malab ucios or bouar value probles. A preliiar esiaio o he soluio is eee o he ierval[ ]. : Reerece o he ucio escribig he ODE as or oexx ucios g: Reerece o a ucio escribig he bouar coiios solii: Preliiar esiaio o he soluio sruc arra. sol: Saplig o he copue soluio sruc arra. See olie help or ore eails