238878905 Metodo de Runge Kutta de 3er Orden en MATLAB

7/21/2019 238878905 Metodo de Runge Kutta de 3er Orden en MATLAB

1/11

25.10 Solucione n

f

ormanumrica el

poblema

siguienle,

de t

=

o a3i

dy

dt

Utilice

el

mtodo

de n(

['orma analtica:

=-y+

y(0)=

1

de

tercer

ordera,

con un tamao de

paso

de o. 5.

*=

-t*r

dy=(-y+tz).rt

bt-tz)dt+dy=o

M=y-t,

u

=

r;i=

o

dM dN

dy

dt

Factor integraDte

dM dN

dlnu

dy dt

dt

dlnu

N

1-O

dtt

dlnu

=

dt

I

or--=l o,

JJ

lnu

=

tlne

Multiplicamos

por

el factor

iDtegrante:

l(y

-

tz).rt

+

dy

=

ol* et

btet

-

t2

et).rt

+

etd.y

=

o

Pilligua Menndez

I-ider Eduardo


2/11

dN

dt

*

2tdt

I

o..

J

t

dt

dM

dy

Io"=

u

du

ye'-

t"e'i-= e'

",,#

=

"

oJ=

"

-

rz.

dl=.r

rlr

'

dy

.rf

dy

I

or=l

o"

J J'

F=yet+7:(t)

df

da-:-

=

yet

+'r'(t)

=

Mf

(x,y)

yet

+

7:'(t)

=

yet

-

t2et

dT

dt

I

ar=-l t'at

JJ

Integracin

por parte

r=-tze,+zlte,t

Integracin

por parte

I-ider Eduardo

illigua Menndez


3/11

Io"=

[*o,

T

=

-tzer

+zte'-zl

e'at

,J

T=-tzer+Zter-zet+c

-yer

=

-tzer

+

Zter

-zer

+

C

Condicin

iricial:

Y(o)

=

r

-(t)eo

= -(o)zeo

+

z(o)eo

-zeo

+

c

-yer

=

-tzer

+

Zter

-zer

+a

yer

=

tzer

-zter

+zer

-

l

Y=t'-zt+2-"'

Verifica

el esultado en MATLAB :

>>

y

=

dsolve

(

'Dy=-y+r"2'

,

'y

(0

)

=1'

,

'r'

)

t^2

-

exp(-t)

-2*L+2

Para

oncontrar

las

soluciones verdaderas

en MATLAB

>>

format long

>>

syms t

>>

y

=

L"2-2*L+2-exp

(-t)

;

>>

t=1nl1ne

(y)

;

Resuftados:

>>

r(0)

1

>>

r(0.5)

Pilligua Menndez Lide Eduardo


4/11


5/11

Para

graficar

en MATLAB:

>>

syms

t

>>

fplot('t"2-2*t+2-exp(-t)

',

t0

31,'r')

>>

qrid

on

RK de

tercer orden

Cor

f(o,1)

K1=

f

(xi,Yi\

f(o,t\=-r+02

Kt=

-a

,11\

Kz=

f

\x+rh,yt+rK)

,1

1

Kz

r(o

+ito.sl,r +it-rlto.sl)

K2

=

f

(o.25,O.7

5\

f

(o.zs,o.

7 s\

=

-o.

7 s

+

(o.zs\2

Kz

=

-O.6A75

Kz=

f(x

+

h,yi

-

K1h+ZKzh\

(3

=

f(o

+

o.

s,1-

(-1Xo.s)

+

2(-o.687sxo.s))

['orma numricr:



6/11

(3

=

f(o.

5, o.812 5)

f(o.

5, o.8125)

=

-o.

8125

+

(o.

s)'z

K:

=

-O.5625

1

yr*,

=

y

*

(Kt

+

4K2

+

K3)h

yos

=

t

+:Ct +

4(-o.6s7s)

+

(-o.

s6zs))(o. s)

Yos

=

o.640625

Con

f(O.

5, O.

64062

5)

Kt

=

f

(x,y)

f

(o.

5,

o.

640625)

=

-o.

640625

+

(o.

s)'z

Kt

=

-O.39O625

x,=

f

(,,+f,n,t,+f,x,n)

x,

=

(o.s

+f,o.s),o.64o62s +;l_o.3e06zs)(o.

s))

K2

= f

(o.

7

s,

o.

s429

6A7

S

)

f

(o.

7

s, o.

s429687

S

)

=

-O.

5429687

5

+

(O.

75)'z

Kz

=

0.01953125

Kz=

f(x

+

h,yi

-

K1h+2Kzh)

h

=

f(o

S+0.5,0.640625

(

0.390625)(0.5)+2(0.01953125)(0.5))

(1

=

f(1,

o.

85546875)

f(1,

o.

85546875)

=

-o.

85546875

+

(1)'z

K3

=

o.14453125

1

yr*,

=

y

*

(Kt

+

4Kz

+

K3)h

1

yl

=

o.640625

+;(

0.390625

+a(0.01953125) +

(0.14453125))(0.5)

y1

=

O.6266276lJ42

Pilligua Menndez Lider Eduado


7/11

Coa

f

(1-,

o.

626627

60

42)

Kt

=

f

(x,y)

f

(1,

o. 626627 6042)

=

-o.

626627 60

42

+

(r)2

K1

=

0.373372395A

x,=

f

(,,+f,n,t,+f,x,n)

x,

=

(r

+lro.

sl,o.6266276r,42

*

|io.

r rrr

rrrr ru)io. ,))

K2

=

f

(1.2

s,

o. 7199

7 07

O32

)

f

(1.

2 S, o. 7 199

7

07

o32

)

=

-o.

7 199

7

07 o32

+

(1.

2

S)2

Kz

=

O.A42529296A

Kz=

f(x

+

h,yi

-

K1h+2Kzh)

&=

(7

+

O.5,O.626627

6042

(O.37

337239 58)

(0.

5)

+

2

(0.

842

5292968)(0.

5

K3

=

f(1.

s,1.2a247o7o3)

f

(1.

S, 1.

282470703)

=

-1.

2s2470703

+

(1.

5)'z

K3

=

0.967

529297

1

yr*,

=

y

*

;(Kt

+

4Kz

+

K3)h

y

I

s

o.h2hh27ho42,

If

o.rrrrrrrr5s

I

4(0.842s292968)

I

(0.967s29297))(0.s)

Y1s

=

l.Ol92l25ll

Coa

f

(1.

S, t.

019212 511)

Kt

= f

(x,y)

f(1.

S, t.otgztzstt)

=

-1..ot92t2st1

+

(1.s)'z

K1

=

1.23lJ747449

x,=

f

(,,*f,o,",*f,*,o)

x,

=

(r.s

+

f,o.

s),t.o:.sz:rzs:.:.

+L(1.

z3o7s74se)

(o.

s))

K2

=

f(1.7

s,1.3269O93A3

)



8/11

f

(1.

7

S, 1. 3269093s3

)

=

-r.

326909383

+

(1.

75)'z

Kz

=

1.735590617

Kz=

f(x

+

h,yi

-

K1h+2Kzh)

I(3

=

f

(1.

5

+

0.

5,

1. 01

s272

577

(7.2307

A7489)(0. 5)

+

2(1. 735590617)(0.

5

K1

=

f

(2,2.

139

4O93A4)

f

(2,

2. 139 409384)

=

-2.

139 409384

+

(2)2

(3

=

1.460590616

1

yr*,

=

y

*

(Kt

+

4K2

+

K3)h

1.

yz

=

7.O7s272577

+

-i(7.nO7A7

4As

+

4(7.735590617)

+

(1.860590616(0.

5)

/z

=

1.4553

57559

Con

f(2,1.

8553 57559)

Kt

=

f

(x,y)

f(2,1.

855357559)

=

-1.855357559

+

(z)'z

K1

= 2.144642441

x,=

f

(,,+ n,t,+ x,n)

x,=

(z

+Lo.

s,1.

ass3

sTsse

*Lr.rnnun

nnrlo. s)

K2

=

f(2.25,2.39151A169

)

f

(2.

2 S, 2. 39 tststg

)

=

-2.

39

rsrs,.g

+

(2.

2 S)2

Kz

=

2.67lJ9AlA3l

Kz=

f(x

+

h,yi

-

K1h+2Kzh)

&= (2

+

o.5,7.855357

SSs

(2.744642441)(0.

5)

+

2(2.670981831)(0.5))

K3

=

f(2.5,3.4S4OtAt7)

f(2.5,3.4s4otst7)

=

-3.4S4o,.st7

+

(2.

S)2

K3

=

2.7959a1a3



9/11

1

yr*,

=

y

*

(Kt

+

4K2

+

K3)h

1.

yzs

=

7.ass3s7

sss

+;(2.744642447 +

4(2.67

osa7a37)

+

(2.7ss98183 (0.

5)

lzs

=

3.1574O3525

Coa

f

(2.

5,3.

1.57

4o3 Sz

S)

Kt

=

f

(x,y)

f

(2.

5,3.

tS7 403525)

=

-3.

157403525

+

(2.

s)'z

K1

=

3.092596475

x,=

f

(,,+f,n,t,+f,x,n)

x,

=

(2.

z

s

+

L

o.

s), 3.

1s7403szs

+

1ir.

orrsrunrs;o.

s;)

K2

=

f(2.7

5,3.930552644

)

f

(2.

7

S, 3.930552644

)

=

-3.

930552

644

+

(2.

7

5)2

Kz

=

3.631947356

Kz=

f(x

+

h,yi

-

K1h+2Kzh)

Kr=

(2.5

+

o.5,3.7574o3525

(3.os25s647

5)(0. 5)

+

2(3.631947356X0.5

K1

=

f

(3,

5.2 43O

526

4 4)

f

(3,

5.

243052644)

=

-

S. 243052644

+

(3)2

K3

=

3.7 56947356

1

yr*,

=

y

*

;(Kt

+

4Kz

+

K3)h

1.

yz

=

3.7s7aBszs

+

i(3.os2ss647

s

+

4(3.637s47 3s6)

+

(3.7

s6s473

56(0.5)

y3

=

4.934447963



10/11

Tabla de resultados con su error verdadero

Programacin

en

MATLAB

:Mtodo

Runge Kutta de

3er orden

clf

v:1;

t:0;

h:0.5;

iter=

(tma

t)

/h;

".Clcuto

de las constantes de Runge kutta

r1:

y +lt^2);

K2:

ly+

lKt/2)

a]n)

+

lt+

ltr) / 12) ) ^2;

K3:

(y (K1*h)+ (2*K2rh)

)+

(t+h)

^2;

y=y+

(

(Kl+

(4

*K2

)

+K3)

/6)

rh;

vectory=[vectory,y];

vectort=[vectort,t];

subplot

(1.1.1)

;

t

v

Eo

o

7

Oo/

o.s

o.6411625

o.

442lJ31943

40/

7

o_ 6266276l)42

o.46497262740/

1.

S

7.O79272577

o.74569614910/

2 1. ass3

s7ss9

o.499133lJ4410/

2.5 3.7574ll3525

o.331A105559%

3

4_934447963

o.2295A544990/



11/11

plot

(wectort.

wectoty,

'

r

*')i

title('Mtodo Runge kutta Tercer orden')

xlabel

('valores

t');

ylabel ('valores y')

;

ResItrdos

cr

MATI,AB

1.000000000000000

o

.

626627 604766667

1

.

855357558638961

4

-

934447963027503

o

-

640625000000000

1- O192

.2

510a50

694

3 _

157O3525011039

0

1.000000000000000

2.000000000000000

3

-

OOOOOOOOOOOOOOO

0. 500000000000000

I

-

500000000000000

2 _ 500000000000000

Grlico

de

la

progrrmacin:


238878905 Metodo de Runge Kutta de 3er Orden en MATLAB

Documents

Transcript of 238878905 Metodo de Runge Kutta de 3er Orden en MATLAB