PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)
-
Upload
martina-mclean -
Category
Documents
-
view
49 -
download
0
description
Transcript of PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)
![Page 1: PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/1.jpg)
PERSAMAAN DIFERENSIAL
(DIFFERENTIAL EQUATION)
metode euler
metode runge-kutta
![Page 2: PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/2.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/3.jpg)
![Page 4: PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/4.jpg)
Rate equations
![Page 5: PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/5.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/6.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/7.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/8.jpg)
PDO
• Dinyatakan dalam 1 peubah dalam menurunkan suatu fungsi
• Contoh:
kPPkPdt
dP
xyxdx
dy
'
sin'sin
![Page 9: PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/9.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/10.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/11.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/12.jpg)
Rate equations
![Page 13: PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/13.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/14.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/15.jpg)
• 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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/16.jpg)
Rate equation State equation(dsolve in Maple)
The Analytical Solution of the Rate Equation is the State Equation
![Page 17: PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/17.jpg)
There are very few models in ecology that can be solved
analytically.
![Page 18: PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/18.jpg)
Solusi Numerik
• Numerical integration– Eulers– Runge-Kutta
![Page 19: PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/19.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/20.jpg)
, 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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/21.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/22.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/23.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/24.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/25.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/26.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/27.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/28.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/29.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/30.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/31.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/32.jpg)
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)](https://reader033.fdocument.pub/reader033/viewer/2022061616/5681300d550346895d9583f3/html5/thumbnails/33.jpg)
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