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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
MtodosdeRungeKutta2orden
Tresecuacionesconcuatroincgnitas,sedebeasumiralgunoelvalordeunavariable.
a1=1a2, p1=q11=
12 a2
ltimaclase 20
MtodosdeRungeKutta2orden
MtododeHeunconunsolocorrector(a2=)
k1= f (x i , yi)k2= f (x i+h , yi+k1h)
yi+1=yi+[ 12 k1+ 12 k2]h
ltimaclase 21
MtodosdeRungeKutta2orden
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
MtodosdeRungeKutta2orden
Elmtododepuntomedio(a2=1)
k1= f (x i , yi)
k2= f (x i+12h , yi+
12k1h)
yi+1=yi+k2 h
ltimaclase 23
MtodosdeRungeKutta2orden
Ejemplo:UtilizarelmtododeRKdesegundogradoparaelpuntomedio,pararesolverlasiguienteecuacindiferencial.
f (x , y)=exp(0.3x)0.3y(xo , yo)=(0 ,0)
ltimaclase 24
MtodosdeRungeKutta2orden
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
MtodosdeRungeKutta3orden
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
MtodosdeRungeKutta4orden
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
MtodosdeRungeKutta4orden
ResolverelproblemaanteriorutilizandoelmtododeRKdecuartogradoycompararlo
conlarespuestamatemtica.
ltimaclase 29
MtodosdeRungeKutta4orden
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