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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.

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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 t0) 0, maka C 0

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)

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

PDO

- Dinyatakan dalam 1 peubah dalam menurunkan suatu

fungsi - Contoh

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)

PDO

Ordiner, linier, orde 3 Ordiner, linier, orde

2 Ordiner, non linier, orde 1

Solusi persamaan diferensial

- Secara analitik, mencari solusi persamaan

diferensial adalah dengan mencari fungsi integral

nya. - Contoh, untuk fungsi pertumbuhan secara

eksponensial, persamaan umum

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- 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

- 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?

Solusi persamaan diferensial

- Jumlah bakteri menjadi dua kali lipat setiap 5

jam, maka k (ln 2)/5 - Jika P0 1 unit, maka setelah satu jam

The Analytical Solution of the Rate Equation is

the State Equation

Rate equation

State equation

(dsolve in Maple)

There are very few models in ecology that can be

solved analytically.

Solusi Numerik

- Numerical integration
- Eulers
- Runge-Kutta

Numerical integration makes use of this

relationship

Which youve 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

Fundamental Approach of Numerical Integration

y f(t), unknown

yt?t, unknown

yt?t, estimated

y

, known

yt, known

?t, specified

t

Calculate dN/dt1 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)

Eulers Method yt ?t yt dy/dt ?t

Example of Numerical Integration

point to estimate

Analytical solution to dy/dt

Y0 10

? t 0.5

Eulers Method yt ?t yt dy/dt ?t

analytical y(t ?t)

m1 dy/dt at yt m1 610-.007(10)2 ?y

m1?t yest yt ?y

y

estimated y(t ?t)

?y

yt 10

? t 0.5

Runge-Kutta Example

point to estimate

Problem estimate the slope to calculate ?y

?y

? t 0.5

Runge-Kutta Example

estimated yt?t

Unknown point to estimate, yt?t

estimated yt?t

estimated yt?t

yt

? t 0.5

Runge-Kutta, 4th order

Uses the derivative, dy/dt, to calculate 4 slopes

(m1m4) within ?t

These 4 slopes are used to calculate a weighted

slope of the state function between t and t ?t,

which is used to estimate yt ?t

Step 1 Evaluate slope at current value of state

variable.

y0 10

m1 dy/dt at y0 m1 610-.007(10)2 m1 59.3

y

m1slope 1

y0

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

624.8-.007(24.8)2 m2 144.63

m2slope 2

y1

? t 0.5/2

Step 3 Calculate y2 at t ?t/2 using

k2. Evaluate slope at y2.

y2 y0 k2 ?t /2 y2 46.2

k3 slope 3

k3 dy/dt at y2 k3 646.2-.007(46.2)2 k3

263.0

y2

? t 0.5/2

Step 4 Calculate y3 at t ?t using k3. Evaluate

slope at y3.

y3 y0 k3 ?t y3 141.5

y3

k4 slope 4

k4 dy/dt at y3 k4 6141.0-.007(141.0)2 k4

706.9

y2

? t 0.5

Now you have 4 calculations of the slope of the

state equation between t and t?t

m4 slope 4

m3 slope3

m2 slope 2

m1 slope 1

? t 0.5

Step 5 Calculate weighted slope. Use weighted

slope to estimate y at t ?t

weighted slope

true value

weighted slope

estimated value

? t 0.5

Conclusions

Analytical

- 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

Eulers

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