06class Mathematica

5/24/2018 06class Mathematica

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How to solve ODEs

using MATHEMATICA

Plasma ApplicationModeling POSTECH

Gan-Young Park and Jae-Koo LeeDepartment of Electronic and Electrical Engineering,POSTECH

2006. 03. 22

EECE490O


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Contents

Basic usages of MATHEMATICA

Solving ODEs

- Eulers method

- Predictor-Corrector method ( second-order )

- Forth-order Runge-Kutta method

- Tridiagonal matrix method


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References

Textbook

- Numerical and Analytical Methods for Scientists and

Engineers Using Mathematica, Daniel Dubin, Wiley, 2003

Web Lecture

: http://www.mathought.com/main.htm
http://www.mathought.com/main.htmhttp://www.mathought.com/main.htm


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Standard Screen

Notebook :working window

( *.nb ) Palettes

( FilePalettes - )

: a collection of numerical expressions or characters


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Basic Usages (1)

Shift + enter: Execution

(shift + enter)

(shift + enter)

In[1]indicates input contents in the line.

Out[1]indicates the result of In[1]

%: a symbol indicating the result obtained right before.


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Basic Usages (2)

application of Palettes

Font size: Format - Size -


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Basic Usages (3)

The names of most of functions included start with a capital letter.

included constants

- Pi- Infinite

- eE

- i = I1

?(function) or ??(function)

: shows the usage of function or options


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Basic Usages (4)

symbol calculation

The symbol * for a product can be replaced by blank.


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Basic Usages (5)

==( equality ), =( substitution ), :=( definition )

Common feature

Different feature


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Plasma Application

Modeling POSTECH

Basic Usages (6)

Common feature

Different feature


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Plasma Application

Modeling POSTECH

Basic Usages (7)

Plot[ function, {variable, a, b}, options] : drawing 2-D graph between a and b


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Plasma Application

Modeling POSTECH

Basic Usages (8)

PlotStyle : a option for coloring

Input - Color Selector


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Basic Usages (9)

PlotLabel : a option for labeling at top of graph

AxesLabel : a option for labeling at axes

AspectRatio : a option for adjusting the ratio of vertical to horizontal

PlotRange : a option for restricting range to plot


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Plasma Application

Modeling POSTECH

Basic Usages (10)

DisplayFunction Identity : a option for not showing a graph, just memorizing it.

DisplayFunction -> $DisplayFunction : a option for showing a graph memorized.

Show : a function to display graphs which have been shown or memorized


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Plasma Application

Modeling POSTECH

Basic Usages (11)

Package

Since path is assigned up to

StandardPackages, just sub-paths

should be written.

The symbol ` is used in

Mathematica instead of \.


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Plasma Application

Modeling POSTECH

Basic Usages (12)

Table [contents, {range of loop}]


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Plasma Application

Modeling POSTECH

Basic Usages (13)

Do [exp, {i,min,max,d}]

For [start, test, i++, body]

While [test, body]

Since Do, For, While dont

show results and save results,

functions such as Print[] should

be used for checking results.

Thats a difference among

Table and above functions.


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Plasma Application

Modeling POSTECH

Basic Usages (14)

Module [ {local variables}, contents ]

: The variables written at {} in Module[ ] are used as local variables which doesnt

affect global variables with same characters and cant be used out of Module[ ].


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Plasma Application

Modeling POSTECH

Solving ODEs

- Eulers method

- Predictor-Corrector method

( second-order )

- Forth-order Runge-Kutta method

- Solving tridiagonal matrix


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Plasma Application

Modeling POSTECH

Eulers method (1)

0

1 1 1

[ , ], (0)

( ) ( ) ( , ( ))n n n n

dvf t v v v

dtv t v t t f t v t


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Eulers method (2)

Plasma Application

Modeling POSTECH


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Predictor-Corrector method ( 2nd-order )

0

1 1 1

1 11

[ , ], (0)

( ) ( ) ( , ( ))

( , ( )) ( , ( ))( ) ( )

2

n n n n

n n n nn n

dvf t v v v

dtv t v t t f t v t

f t v t f t v tv t v t t


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Plasma Application

Modeling POSTECH

Forth-order Runge-Kutta method

0

1 1 1

12 1 1

23 1 1

4 1 1 3

1 1 2 3 4

[ , ], (0)

( , ( ))

( , ( ) )2 2

( , ( ) )2 2

( , ( ) )

1( ) ( ) [ 2 2 ]

6

n n

n n

n n

n n

n n

dvf t v v v

dt

k t f t v t

ktk t f t v t

ktk t f t v t

k t f t t v t k

v t v t k k k k

( based on the Simpsons 1/3 rule)


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Example 9.1 ( Nakamura )

20 0.5 20

' 20 7exp( 0.5 ), y(0) 5,

. : ( ) 5 (7 19.5)( )t t t

y y t

exact sol y t e e e


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Program 9-1 ( Nakamura )

0)()()(2

2

tkytydt

dBtydt

dM , y(0)=1, y(0)=0

Changing 2ndorder ODE to 1storder ODE,

)(),,()( tztzyftydt

d

0)()()( tkytBztzdt

dM

)()(),,()( tyM

ktzM

Btzygtzdt

d

(1)

(2)

y(0)=1

z(0)=0

Plasma Application

Modeling POSTECH


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Plasma Application

Modeling POSTECH

To solve 2nd-order ODE using second-order Runge-Kutta method


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Plasma Application

Modeling POSTECH

1)(*)(' tytty y(0)=0)(

)()('

22tyt

tyty

y(0)=1

To solve ODE using fourth-order Runge-Kutta method

P 10 1 ( N k )


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)2.0exp()()(2 xxyxy

)10()10(,1)0( yyy

Solve difference equation,

With the boundary conditions,

x = 0 1 2i = 0 1 2

9 109 10

112

11 22

)(

iii

iii yyyh

yyyxy

)2.0exp()2(2 11 iyyyy iiii )2.0exp()()(2 xxyxy

1)0( yEspecially for i = 1, 2)2.0exp(25 21 yy

known

y(0)=1

)10()10( yy

Plasma Application

Modeling POSTECH


P 10 1 ( N k )


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For i = 10,

1091110

911

10 22| yyyy

yy

y

)10()10( yy

)2exp(252 11109 yyy )2exp(5.05.42 109 yy

Summarizing the difference equations obtained, we write

)2.0exp(252 11 iiii xyyy 2)2.0exp(25 21

yy

)2exp(5.05.42 109 yy

)2exp(5.0

)32.0exp(

)22.0exp(

2)2.0exp(

5.42

252

252

25

10

3

2

1

y

y

y

y

Tridiagonal matrix



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Solution Algorithm for Tridiagonal Equations (1)

3

2

1

3

2

1

33

222

11

0

0

D

D

D

BA

CBA

CB

3

2

1

1

2

3

2

1

33

222

1

1

21

1

2

0

0

D

D

DB

A

BA

CBA

CB

AB

B

A

R2

3

2

12

3

2

1

33

222

122

0

0

D

D

DR

BA

CBA

CRA

3

122

12

3

2

1

33

2122

122

0

0

0

D

DRD

DR

BA

CCRB

CRA

2B 2D

3

2

2

3

12

3

2

1

33

2

2

32

2

3

122

0

0

0

D

DB

A

DR

BA

CB

AB

B

A

CRA

R3

233

23

12

3

2

1

233

233

122

00

0

0

DRD

DR

DR

CRB

CRA

CRA

3B 3D

Based on Gauss elimination

Plasma Application

Modeling POSTECH


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3

23

12

3

2

1

3

233

122

00

0

0

D

DR

DR

B

CRA

CRA

3

33

B

D

2

33322

3

322332323 ,)(

B

ARCD

A

RDRCRA

)(1 3222

2 CDB

1

22211

2

211221212 ,)(

B

ARCD

A

RDRCRA

)(1

)(1

211

1

211

1

1 CDB

CDB

Solution Algorithm for Tridiagonal Equations (2)

Plasma Application

Modeling POSTECH

( k )


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Plasma Application

Modeling POSTECH


To solve boundary-value problem

using tridiagonal method

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