View by Category

Loading...

PPT – PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION) PowerPoint presentation | free to download - id: 3cfaaf-MWI4O

The Adobe Flash plugin is needed to view this content

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

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.

(No Transcript)

(No Transcript)

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

(No Transcript)

- 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

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Home About Us Terms and Conditions Privacy Policy Contact Us Send Us Feedback

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2017 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "PERSAMAAN DIFERENSIAL (DIFFERENTIAL EQUATION)" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!