2. 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.
50=
dt
dx
5. 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
6. 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)
7. PDO
Pers.dif. Ordiner = pers. yg mengandung
sejumlah tertentu turunan ordiner dari satu
atau lebih variabel tak bebas terhadap satu
variabel bebas.
y(t) = variabel tak bebas
t = variabel bebas
dan turunan y(t)
Pers di atas: ordiner, orde dua, linier
t
ety
dt
tdy
t
dt
tyd
=++ )(5
)()(
2
2
8. PDO
• Dinyatakan dalam 1 peubah dalam
menurunkan suatu fungsi
• Contoh:
kPPkP
dt
dP
xyx
dx
dy
=>−=
=>−=
'
sin'sin
9. 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
δ
δ
δ
δ
11. Solusi persamaan diferensial
• Secara analitik, mencari solusi persamaan
diferensial adalah dengan mencari fungsi
integral nya.
• Contoh, untuk fungsi pertumbuhan secara
eksponensial, persamaan umum:
kP
dt
dP
=
13. 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
14. 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?
15. • 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
kP
dt
dP
=
dtk
P
dP
t
t
P
P
∫∫ =
1
0
1
0
)(ln 0
0
ttCk
P
P
−=
kt
ePtP 0)( =
)(1)1(
)1)(
5
)2(ln(
eP =
1487.1=
16. Rate equation State equation(dsolve in Maple)
The Analytical Solution of the Rate Equation
is the State Equation
17. There are very few models in
ecology that can be solved
analytically.
19. 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:
1where,11 =∆∆
−+=+ tt
K
N
rNNN t
ttt
t
dt
dy
yy ttt ∆+≈∆+
20. , known
Fundamental Approach of Numerical Integration
y = f(t), unknown
∆t, specified
y
t
yt, known
dt
dy
yt+∆t, estimated
t
dt
dy
yy ttt ∆+≈∆+
yt+∆t,
unknown
21. Euler’s Method: yt+ ∆t ≈ yt + dy
/dt ∆t
1where,1 =∆∆
−+=∆+ tt
K
N
rNNN t
tttt
dt
dN
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
22. Example of Numerical Integration
dy
dt
y y= −6 007 2
.
Analytical solution to dy/dt
Y0 = 10
∆ t = 0.5
point to
estimate
23. 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
dt
y y= −6 007 2
.
26. 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
∆+∆+=
∆+∆+′=
∆+∆+′=
′=
),(atderivative),( ytytf =′
tmmmmyy ttt ∆++++=∆+ )22(
6
1
4321
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:
27. 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 = 59.3
m1=slope 1
y0
28. 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
29. 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
30. 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
31. 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
32. Step 5:
Calculate weighted slope.
Use weighted slope to estimate y at t +∆t
∆ t = 0.5
weighted slope =
true value
estimated value
weighted slope
tmmmmyy ttt ∆++++=∆+ )22(
6
1
4321
)22(
6
1
4321 mmmm +++
33. 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
Editor's Notes
When you make a model in ecology, what usually happens is you make a box and arrow diagram
And then decide how each of those boxes change…that is, the rates of input and output into and out of each box.
And what you end up with is a bunch of rate equations
And then decide how each of those boxes change…that is, the rates of input and output into and out of each box.
And what you end up with is a bunch of rate equations
If you are working with simple equations, such as the logistic, you can solve your known rate equation analytically to find the state equation
and see how N changes over time with different parameter values
In most cases your equation or system of equations will be too complicated to find an analytical solution.
If this case we have to resort to numerical techniques to estimate the state equation
Which estimates y(t + delta-t) by adding the rate equation dy/dt * change in time to the original, specified, value of y(t).
Which you’ve seen before, primarily as the relationship between discrete and continuous time models…
This relationship is the basis of numerical integration – that is using your known rate equation, dy/dt, to calculate the slope at y at t, which is specified, to estimate y at t+delta-t.
The basic problem is that you don’t know the state equation but you want to know how y changes through time as a result of your known rate equation dy/dt. So you use the rate equation at a known point, here y at t, to find the slope of the state equation (or dy/dt) at y at t. You then use this slope, over the specified time interval, delta-t, to estimate the next point along the unknown state equation. In this case, that next point is y(t + delta-t)
As I mentioned before, this is what you did when you used the logisitic rate equation to estimate points along the state function at a constant t step over a long time interval,
so what I want to point out here is that this it is an iterative process.
This method of numerical integration is called euler’s method, and is the simplest form of numerical integration. When delta t is small it works reasonably well but it is not always the best choice for estimating the state equation.
And I will use this simple differential equation, for which there is a solution to demonstrate this idea.
The main goal of any num int technique is to accurately estimate points along the unknown state function using the rate equation.
Remember that we don’t really know the line, we only know its derivative. Line is there for illustration only.
The line is the function whose derivative is given above.
In many cases, when delta t is not very small, euler’s method does not give a very accurate or satisfying estimate of the true y (t+delta-t)
because the slope at y at t does not estimate the change in y over delta-t accurately
The challenge, then, is finding a slope that will more accurately estimate the change in y over delta-t to get a more accurate estimate of the point along the unknown state function – this is what runga kutta attempts to do.
Remember that we don’t really know the line, we only know its derivative. Line is there for illustration only.
The line is the function whose derivative is given above.
If you have a relatively large delta-t and the slope of the unknown state function is not linear between t and t+delta t, the slope of the line may vary dramatically between t and t+delta-t
So that using a single calculation of the slope of the state function at time between t and t + delta-t to estimate y(t+delta-t) might not give you a very good estimate the true value
The fourth order runga kutta method is the most frequently used technique of numerical integration – it uses four calculations of the slope within the specified time step to more accurately represent the change in y, and therefore give a more accurate estimate of y at t+delta-t.
Slope is calculated four times at t, ½ delta-t, and delta-t
This slope is then used to calculate y1, which is the point along that line at ½ delta-t
Dy/dt is then used to calculate the slope at y1
Now you have 4 slopes, some of which are much steeper than the slope that would be needed to calculate f(t+delta-t) and some that are much less steep.
All of these are used to calculate a weighted average slope of the state equation between t and t + delta t that is used to estimate y(t+delta-t)
which results in a much more accurate estimate than techniques such as eulers.