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PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION) metode euler metode runge-kutta

PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)

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PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION). metode euler metode runge-kutta. Persamaan Diferensial. Persamaan paling penting dalam bidang rekayasa, paling bisa menjelaskan apa yang terjadi dalam sistem fisik. Menghitung jarak terhadap waktu dengan kecepatan tertentu, 50 misalnya. - PowerPoint PPT Presentation

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Page 1: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

PERSAMAAN DIFERENSIAL

(DIFFERENTIAL EQUATION)

metode euler

metode runge-kutta

Page 2: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Persamaan Diferensial

• Persamaan paling penting dalam bidang rekayasa, paling bisa menjelaskan apa yang terjadi dalam sistem fisik.

• Menghitung jarak terhadap waktu dengan kecepatan tertentu, 50 misalnya.

50dt

dx

Page 3: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)
Page 4: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Rate equations

Page 5: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Persamaan Diferensial

• Solusinya, secara analitik dengan integral,

• C adalah konstanta integrasi

• Artinya, solusi analitis tersebut terdiri dari banyak ‘alternatif’

• C hanya bisa dicari jika mengetahui nilai x dan t. Sehingga, untuk contoh di atas, jika x(0) = (x saat t=0) = 0, maka C = 0

dtdx 50 Ctx 50

Page 6: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Klasifikasi Persamaan Diferensial

Persamaan yang mengandung turunan dari satu atau lebih variabel tak bebas, terhadap satu atau lebih variabel bebas.

• Dibedakan menurut:– Tipe (ordiner/biasa atau parsial)– Orde (ditentukan oleh turunan tertinggi yang

ada– Liniarity (linier atau non-linier)

Page 7: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

PDOPers.dif. Ordiner = pers. yg mengandung sejumlah tertentu turunan ordiner dari satu atau lebih variabel tak bebas terhadap satu variabel bebas.

y(t) = variabel tak bebast = variabel bebasdan turunan y(t)

Pers di atas: ordiner, orde dua, linier

tetydt

tdyt

dt

tyd )(5

)()(2

2

Page 8: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

PDO

• Dinyatakan dalam 1 peubah dalam menurunkan suatu fungsi

• Contoh:

kPPkPdt

dP

xyxdx

dy

'

sin'sin

Page 9: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Partial Differential Equation• Jika dinyatakan dalam lebih dari 1 peubah, disebut

sebagai persamaan diferensial parsial• Pers.dif. Parsial mengandung sejumlah tertentu

turunan dari paling tidak satu variabel tak bebas terhadap lebih dari satu variabel bebas.

• Banyak ditemui dalam persamaan transfer polutan (adveksi, dispersi, diffusi)

0),(),(

2

2

2

2

t

txy

x

txy

Page 10: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

PDO

xeyy

dt

sd

yy

3)'(

32

24'''

2

2

2

Ordiner, linier, orde 3

Ordiner, linier, orde 2

Ordiner, non linier, orde 1

Page 11: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Solusi persamaan diferensial

• Secara analitik, mencari solusi persamaan diferensial adalah dengan mencari fungsi integral nya.

• Contoh, untuk fungsi pertumbuhan secara eksponensial, persamaan umum:

kPdt

dP

Page 12: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Rate equations

Page 13: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

But what you really want to know is…

the sizes of the boxes (or state variables) and how they change through time

That is, you want to know:

the state equations

There are two basic ways of finding the state equations for the state variables based on your known rate equations:

1) Analytical integration 2) Numerical integration

Page 14: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Suatu kultur bakteria tumbuh dengan kecepatan yang proporsional dengan jumlah bakteria yang ada pada setiap waktu. Diketahui bahwa jumlah bakteri bertambah menjadi dua kali lipat setiap 5 jam. Jika kultur tersebut berjumlah satu unit pada saat t = 0, berapa kira-kira jumlah bakteri setelah satu jam?

Page 15: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

• Jumlah bakteri menjadi dua kali lipat setiap 5 jam, maka k = (ln 2)/5

• Jika P0 = 1 unit, maka setelah satu jam…

Solusi persamaan diferensial

kPdt

dP

dtkP

dPt

t

P

P 1

0

1

0

)(ln 00

ttCkP

P

ktePtP 0)(

)(1)1()1)(5

)2(ln(eP

1487.1

Page 16: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Rate equation State equation(dsolve in Maple)

The Analytical Solution of the Rate Equation is the State Equation

Page 17: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

There are very few models in ecology that can be solved

analytically.

Page 18: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Solusi Numerik

• Numerical integration– Eulers– Runge-Kutta

Page 19: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Numerical integration makes use of this relationship:

Which you’ve seen before…

Relationship between continuous and discrete time models

*You used this relationship in Lab 1 to program the

logistic rate equation in Visual Basic:

1 where,11

tt

K

NrNNN t

ttt

tdt

dyyy ttt

Page 20: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

, known

Fundamental Approach of Numerical Integration

y = f(t), unknown

t, specified

y

t

yt, knowndt

dy

yt+t, estimated

tdt

dyyy ttt

yt+t,

unknown

Page 21: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Euler’s Method: yt+ t ≈ yt + dy/dt t

1 where,1

tt

K

NrNNN t

tttt

dtdN

Calculate dN/dt*1 at Nt

Add it to Nt to estimate Nt+ t

Nt+ t becomes the new Nt

Calculte dN/dt * 1 at new Nt

Use dN/dt to estimate next Nt+ t

Repeat these steps to estimate the state function over your desired time length

(here 30 years)

Nt/K with time, lambda = 1.7, time step = 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 10 20 30 40 50

time (years)

Nt/K

Page 22: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Example of Numerical Integrationdy

dty y 6 007 2.

Analytical solution to dy/dt

Y0 = 10

t = 0.5

point to estimate

Page 23: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

y

Euler’s Method: yt+ t ≈ yt + dy/dt t

yt = 10

m1 = dy/dt at yt

m1 = 6*10-.007*(10)2

y = m1*t

yest= yt + y

t = 0.5

y

estimated y(t+ t)

analytical y(t+ t)

dy

dty y 6 007 2.

Page 24: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Runge-Kutta Exampledy

dty y 6 007 2.

t = 0.5

point to estimate

Problem: estimate the slope to

calculate y

y

Page 25: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Runge-Kutta Example

yt

Weighted

avera

ge of >

1 slope

Unknown point to estimate, yt+Δt

½ Δt Δt t

estimated yt+Δt

estimated yt+Δt

estimated yt+Δt

t = 0.5

Page 26: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Uses the derivative, dy/dt, to calculate 4 slopes (m1…m4) within Δt:

Runge-Kutta, 4th order

),(

)2/,2/(

)2/,2/(

),(

34

23

12

1

tmyttm

tmyttfm

tmyttfm

ytfm

),(at derivative),( ytytf

tmmmmyy ttt )22(6

14321

These 4 slopes are used to calculate a weighted slope of the state function between t and t + Δt, which is used to estimate yt+ Δt:

Page 27: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

y

Step 1:

Evaluate slope at current value of state variable.

y0 = 10

m1 = dy/dt at y0

m1 = 6*10-.007*(10)2

m1 = 59.3m1=slope 1

y0

Page 28: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Step 2:

A) Calculate y1at t +t/2 using m1.

B) Evaluate slope at y1.

A) y1 = y0 + m1* t /2

y1 = 24.82

B) m2 = dy/dt at y1

m2 = 6*24.8-.007*(24.8)2

m2 = 144.63 m2=slope 2

t = 0.5/2

y1

Page 29: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Step 3:

Calculate y2 at t +t/2 using k2.

Evaluate slope at y2.

y2 = y0 + k2* t /2

y2 = 46.2

k3 = dy/dt at y2

k3 = 6*46.2-.007*(46.2)2

k3 = 263.0

k3 = slope 3

t = 0.5/2

y2

Page 30: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Step 4:

Calculate y3 at t +t using k3.

Evaluate slope at y3.

y3 = y0 + k3* t

y3 =141.5

k4 = dy/dt at y3

k4 = 6*141.0-.007*(141.0)2

k4 = 706.9

k4 = slope 4

t = 0.5

y2

y3

Page 31: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

m4 = slope 4

t = 0.5

m3 = slope3

m2 = slope 2

m1 = slope 1

Now you have 4 calculations of the slope of the state equation between t and t+Δt

Page 32: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Step 5:

Calculate weighted slope.

Use weighted slope to estimate y at t +t

t = 0.5

weighted slope =

true value

estimated valueweighted slope

tmmmmyy ttt )22(6

14321

)22(6

14321 mmmm

Page 33: PERSAMAAN  DIFERENSIAL (DIFFERENTIAL EQUATION)

Conclusions

• 4th order Runge-Kutta offers substantial improvement over Eulers.

• Both techniques provide estimates, not “true” values.

• The accuracy of the estimate depends on the size of the step used in the algorithm.

Runge-Kutta

Analytical

Eulers