unidad 06 ecuaciones diferenciales imprimir

EcuacionesDiferenciales

UniversidaddeElSalvadorFacultaddeIngenierayArquitectura

EscueladeIngenieraElctrica

AnlisisNumrico

WilberCaldern

MtododeEuler

AnchodelintervaloEDO

ValoranteriorNuevaaproximacin

dydx

= f x , y= x i , yi=

yi1=yi xi , yi h

MtododeEuler

Seaf(x,y)=2y(xo,yo)=(1,4)encontrarlasolucinverdaderaycompararlaconelmtododeEuler

i xi Euler Solucin

0 -1.00 4.00 4.00

1 -0.75 4.50 4.50

2 -0.50 5.00 5.00

3 -0.25 5.50 5.50

4 0.00 6.00 6.00

5 0.25 6.50 6.50

6 0.50 7.00 7.00

7 0.75 7.50 7.50

8 1.00 8.00 8.00

MtododeEuler

Seaf(x,y)=6x,y(xo,yo)=(0,2)encontrarlasolucinverdaderaycompararlaconelmtododeEuler

h=0.5 h=0.05 h=0.0005

i xi Solucin Euler Euler Euler

0 0 2.00 2.00 2.00 2.00

1 0.5 2.75 2.00 2.68 2.75

2 1 5.00 3.50 4.85 5.00

3 1.5 8.75 6.50 8.53 8.75

4 2 14.00 11.00 13.70 14.00

5 2.5 20.75 17.00 20.38 20.75

6 3 29.00 24.50 28.55 29.00

7 3.5 38.75 33.50 38.23 38.74

8 4 50.00 44.00 49.40 49.99

MtododeHeun

yi+10 =yi+ f (x i , yi) h

yi+1=yi+f (x i , yi)+ f (xi+1 , yi+1

0 )2

h

yi+10 :Predictor

yi+1 :Corrector

MtododeHeun

Seaf(x,y)=6x,y(xo,yo)=(0,2)encontrarlasolucinverdaderaycompararlaconelmtododeEuler

i xi Solucin Predictor Corrector

0 0.0 2.00 2.00 2.00

1 0.5 2.75 2.00 2.75

2 1.0 5.00 4.25 5.00

3 1.5 8.75 8.00 8.75

4 2.0 14.00 13.25 14.00

5 2.5 20.75 20.00 20.75

6 3.0 29.00 28.25 29.00

7 3.5 38.75 38.00 38.75

8 4.0 50.00 49.25 50.00

MtododeHeun

yi+10 =yi+ f (xi , yi) h

yi+11 =yi+

f (x i , yi)+ f (x i+1 , yi+10 )

2h

yi+12 =yi+

f (x i , yi)+ f (x i+1 , yi+11 )

2h

yi+13 =yi+

f (x i , yi)+ f (x i+1 , yi+12 )

2h

yi+1n1=yi+

f (xi , yi)+ f (xi+1 , yi+1n2)

2h

yi+1n =yi+

f (x i , yi)+ f (xi+1 , yi+1n1)

2h

MtododeHeun

Aplicar el mtodo de Heun a la siguiente ecuacindiferencialordinaria.a)Paradoscorrectoresb)ParacuatrocorrectoresPresentarlasolucinhastax=2enavancesde0.25

=e0.3 x0.3 y(xo , yo)=(0,0)

Paradoscorrectores

.

i x Predictor Corrector 1 Corrector 2

0 0 0 0 0

1 0.25 0.25 0.23159 0.23228

2 0.50 0.4468 0.43037 0.43099

3 0.75 0.61384 0.59921 0.59976

4 1.00 0.75441 0.7414 0.74188

5 1.25 0.87145 0.8599 0.86033

6 1.50 0.96763 0.9574 0.95778

7 1.75 1.04535 1.03631 1.03665

8 2.00 1.10679 1.09882 1.09912

Paratrescorrectores

.

i x Predictor Corrector 1 Corrector 2 Corrector 3

0 0 0 0 0 0

1 0.25 0.25 0.23159 0.23228 0.23226

2 0.50 0.44677 0.43035 0.43097 0.43094

3 0.75 0.6138 0.59917 0.59972 0.5997

4 1.00 0.75435 0.74134 0.74182 0.74181

5 1.25 0.87138 0.85983 0.86026 0.86024

6 1.50 0.96755 0.95731 0.9577 0.95768

7 1.75 1.04526 1.03622 1.03656 1.03655

8 2.00 1.1067 1.09872 1.09902 1.09901

Paracuatrocorrectores

.

i x Predictor Corrector 1 Corrector 2 Corrector 3 Corrector 4

0 0 0 0 0 0 0

1 0.25 0.25 0.23159 0.23228 0.23226 0.23226

2 0.50 0.44677 0.43035 0.43097 0.43094 0.43094

3 0.75 0.6138 0.59917 0.59972 0.5997 0.5997

4 1.00 0.75435 0.74134 0.74183 0.74181 0.74181

5 1.25 0.87138 0.85983 0.86026 0.86024 0.86025

6 1.50 0.96755 0.95732 0.9577 0.95769 0.95769

7 1.75 1.04527 1.03622 1.03656 1.03655 1.03655

8 2.00 1.1067 1.09873 1.09902 1.09901 1.09901

Importante

NoolvidarqueelmtododeHeunsellenaporfila,ylacalculadoradebeestar

enradianes.

Importante

ParaqueelmtododeHeundemejoresresultadossedebencalcularvarioscorrectoresyutilizarpequeosavances.

MtododePuntoMedio

Estudiarparaantesdeexamen.

ltimaclase 15

MtodosdeRungeKutta

(xi,yi,h):funcinincremento

yi1=yi xi , yi , h h

xi , yi , h=a1k1a2 k2a3k3...an kn

ltimaclase 16

MtodosdeRungeKutta

asonconstantesylaskk1= f x i , yi

k2= f x ip1h , yiq11 k1hk3= f x ip2 h , yiq21 k1hq22 k2 h

k n= f x i pn1h , yiqn1,1k1hqn1,2 k2 hqn1,n1k n1h

ltimaclase 17

MtodosdeRungeKutta

Observequelasktienenrelacionesderecurrencia.Estoes,k1,apareceenlaecuacinparacalculark2,lacual

apareceenlaecuacindek3,etctera.

ltimaclase 18

MtodosdeRungeKutta2orden

Donde

yi1=yia1k1a2 k2 h

k1= f x i , yik2= f x i p1h , yiq11 k1h

ltimaclase 19


Tresecuacionesconcuatroincgnitas,sedebeasumiralgunoelvalordeunavariable.

a1=1a2, p1=q11=

12 a2

ltimaclase 20


MtododeHeunconunsolocorrector(a2=)

k1= f (x i , yi)k2= f (x i+h , yi+k1h)

yi+1=yi+[ 12 k1+ 12 k2]h

ltimaclase 21


MtodoRalston(a2=2/3)

k1= f (x i , yi) k2= f (xi+ 34 h , yi+ 34 k1h)yi+1=yi+[ 13 k1+ 23 k2]h

ltimaclase 22


Elmtododepuntomedio(a2=1)

k1= f (x i , yi)

k2= f (x i+12h , yi+

12k1h)

yi+1=yi+k2 h

ltimaclase 23


Ejemplo:UtilizarelmtododeRKdesegundogradoparaelpuntomedio,pararesolverlasiguienteecuacindiferencial.

f (x , y)=exp(0.3x)0.3y(xo , yo)=(0 ,0)

ltimaclase 24


i x k1 k2 y

0 0.00 - - 0

1 0.25 1 0.92569 0.23142

2 0.50 0.85832 0.79198 0.42942

3 0.75 0.73188 0.67276 0.59761

4 1.00 0.61923 0.56662 0.73926

5 1.25 0.51904 0.47231 0.85734

6 1.50 0.43009 0.38866 0.95451

7 1.75 0.35128 0.31463 1.03317

8 2.00 0.28161 0.24927 1.09548

ltimaclase 25


k1= f (x i , yi)

k2= f ( x i+ 12 h , yi+ 12 k1h)k3= f (x i+h , y ik1h+2k2 h)

y i+1=yi+16 [k1+4 k2+k3 ]h

ltimaclase 26

function[x,ysol]=rk3(xini,yini,xfin,h,dydx)

fxy=inline(dydx,"x","y")x=xini:h:xfin;ysol=zeros(1,length(x));,ysol(1)=yini;forp=1:length(x)1

k1=fxy(x(p),ysol(k));k2=fxy(x(p)+0.5*h,ysol(p)+0.5*k1*h);k3=fxy(x(p)+h,ysol(p)k1*h+2*k2*h);

ysol(p+1)=ysol(p)+h*(k1+4*k2+k3)/6;endforendfunction

k1= f (x i , yi) k2= f ( xi+ 12 h , yi+ 12 k1h)k3= f (x i+h , yik1h+2k2 h )

yi+1=yi+16 [k1+4 k 2+k3 ]h

ltimaclase 27


k1= f ( x i , yi) , k2= f ( x i+ 12 h , yi+ 12 k1h)k3= f (x i+ 12 h , yi+ 12 k 2h)k 4= f ( x i+h , yi+k3h )

yk+1=yi+16(k1+2k2+2k3+k 4)h

ltimaclase 28


ResolverelproblemaanteriorutilizandoelmtododeRKdecuartogradoycompararlo

conlarespuestamatemtica.

ltimaclase 29


i x k1 k2 k3 k4 y

0 0.00 - - - - 0

1 0.25 1 0.92569 0.92848 0.85811 0.23194

2 0.50 0.85816 0.79184 0.79432 0.73155 0.43035

3 0.75 0.7316 0.67249 0.6747 0.61881 0.59889

4 1.00 0.61885 0.56625 0.56823 0.51854 0.74082

5 1.25 0.51857 0.47186 0.47361 0.42952 0.85911

6 1.50 0.42956 0.38815 0.3897 0.35067 0.95644

7 1.75 0.3507 0.31408 0.31545 0.28096 1.03522

8 2.00 0.28099 0.24868 0.24989 0.2195 1.09762

ltimaclase 30

ltimaclase 31

PREGUNTAS?

ltimaclase 32

Tarea

Entrega:23dejuniode8:00a10:00a.m.enlaoficinadelprofesor.

ltimaclase 33

FINDEUNIDAD

FINDELCONTENIDO

Diapositiva 1Diapositiva 2Diapositiva 3Diapositiva 4Diapositiva 5Diapositiva 6Diapositiva 7Diapositiva 8Diapositiva 9Diapositiva 10Diapositiva 11Diapositiva 12Diapositiva 13Diapositiva 14Diapositiva 15Diapositiva 16Diapositiva 17Diapositiva 18Diapositiva 19Diapositiva 20Diapositiva 21Diapositiva 22Diapositiva 23Diapositiva 24Diapositiva 25Diapositiva 26Diapositiva 27Diapositiva 28Diapositiva 29Diapositiva 30Diapositiva 31Diapositiva 32Diapositiva 33

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