mws gen ode ppt shooting

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01/19/15 http://numericalmethods.eng.usf.edu 1

Shooting Method

Major: All Engineering Majors

Authors: Autar Ka! "harlie #ar$er

http://numericalmethods.eng.usf.edu%ransforming &umerical Methods Education for S%EM

'ndergraduates
http://numericalmethods.eng.usf.edu/http://numericalmethods.eng.usf.edu/http://numericalmethods.eng.usf.edu/http://numericalmethods.eng.usf.edu/


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Shooting Method

http://numericalmethods.eng.usf.edu
http://numericalmethods.eng.usf.edu/http://numericalmethods.eng.usf.edu/http://numericalmethods.eng.usf.edu/http://numericalmethods.eng.usf.edu/


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http://numericalmethods.eng.usf.edu(

Shooting Method

%he shooting method uses the methods used in sol)ing initial )aluepro*lems. %his is done *+ assuming initial )alues that ould ha)e

*een gi)en if the ordinar+ di,erential e-uation ere a initial )aluepro*lem. %he *oundar+ )alue o*tained is compared ith the actual*oundar+ )alue. 'sing trial and error or some scientic approach!one tries to get as close to the *oundar+ )alue as possi*le.


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Eample

ba

r

( )

( ) 0030770.08

,0038731.05

,01

22

2

=

=

=+

u

u

r

u

dr

du

rdr

ud

et

wdr

du=

%hen

01

2 =+

r

uw

rdr

dw

2here a 3 5

and * 3 4


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http://numericalmethods.eng.usf.edu5

Solution%o rst order di,erential e-uations are gi)en as

( ) 0038371.05, == uwdr

du

( ) knownnotwr

u

r

w

dr

dw5,

2 =+=

et us assume

( ) ( ) ( ) ( )

00026538.058

5855 =

=

uu

dr

duw

%o set up initial )alue pro*lem

( ) ( ) 0038371.05,,,1 === uwurfwdr

du

( ) ( ) 00026538.05,,,22

==+= wwurfr

u

r

w

dr

dw


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Solution "ont'sing Euler6s method!

( )hwurfuu iiiii ,,11 +=+

( )hwurfww iiiii ,,21 +=+

et us consider segments *eteen the to *oundaries!

and then!

5=r

8=

r

75.04

58=

=h


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Solution "ont00026538.0,0038371.0,5,0

000 ==== wuri8or

( )

( )( )

( ) ( )

0036741.0

75.000026538.00038371.0

75.000026538.0,0038371.0,50038371.0

,,

1

000101

=

+=

+=

+=

f

hwurfuu

( )

( )

( )

00010938.0

75.05

0038371.0

5

00026538.000026538.0

75.0)00026538.0,0038371.0,5(00026538.0

,,

2

2

000201

=

+

+=

+=+=

f

hwurfww


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Solution "ont8or ,75.575.05,1 01 =+=+== hrri 00010940.0,0036741.0 11 == wu

( )

( )( )( )( )

0035920.0

75.000010938.00036741.0

75.000010938.0,0036741.0,75.50036741.0

,,

1

111112

=

+=

+=

+=

f

hwurfuu

( )

( )( )

( ) ( )

000011769.0

75.000013015.000010938.0

75.000010938.0,0036741.0,75.500010938.0

,,

2

111212

=

+=

+=

+=

f

hwurfww


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Solution "ont8or 5.675.075.5,2 12 =+=+== hrri 000011785.0,0035920.0 22 == wu

( )

( )( )

( ) ( )

0035832.0

75.0000011769.00035920.0

75.0000011769.0,0035920.0,5.60035920.0

,,

1

222123

=

+=

+=

+=

f

hwurfuu

( )

( ) ( )

( )( )

000053352.0

75.0000086829.0000011769.0

75.0000011769.0,0035920.0,5.6000011769.0

,,

2

222223

=

+=

+=

+=

f

hwurfww


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Solution "ont25.775.050.6,3 23 =+=+== hrri 000053332.0,0035832.0 33 == wu8or

( )

( )( )

( )( )0036232.0

75.0000053352.00035832.0

75.0000053352.0,0035832.0,25.70035832.0

,,

1

333134

=

+=

+=

+=

f

hwurfuu

( )

( ) ( )

( )( )000098961.0

75.0000060811.0000053352.0

75.0000053352.0,0035832.0,75.5000011785.0

,,

2

333234

=

+=

+=

+=

f

hwurfww

875.025.734 =+=+== hrrr

( ) 0036232.084 = uu

So at


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Solution "ontet us assume a ne )alue for( )5

dr

du

( ) ( ) ( ) ( )

( ) 00053076.000026538.025858

2525 ==

=

uu

dr

duw

'sing 75.0=h and Euler6s method! e get

( ) "0029665.084 = uu

2hile the gi)en )alue of this *oundar+ condition is

( ) 0030770.084 = uu


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Solution "ont'sing linear interpolation on the o*tained data for the to assumed )al

( )5dr

du e get

( ) 00030770.08 =u

( ) ( )

( ) ( )00026538.00036232.00030770.00036232.00029645.0

00026538.000053076.05 +

dr

du

00048611.0=

'sing 75.0=h and repeating the Euler6s method ith 00048611.0)5( =w

( ) 0030769.084 = uu


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"omparisons of di,erent

initial guesses

" i f E l d


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"omparison of Euler andunge;Kutta esults ith

eact resultsTable 1 "omparison of Euler and unge;Kutta results ith eact resul

r 10?(

(.0770>10?(

(.47(1>10?(

(.5045>10?(

(.454>10?(

(.1514>10?(

(.0770>10?(

0.00001.(7(11.54

9.497>10?1

1.9500>10?(

(.47(1>10?(

(.555>10?(

(.((1>10?(

(.179>10?(

(.07(>10?(

0.0000(.54>10?

7.0(7>10?

1.11>10?1

1.514>10?1

%t %t


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Additional esources8or all resources on this topic such as digitalaudio)isual lectures! primers! tet*oo$ chapters!multiple;choice tests! or$sheets in MA%A#!

MA%@EMA%"A! Math"ad and MABE! *logs!related ph+sical pro*lems! please )isit

http://numericalmethods.eng.usf.edu/topics/sho

otingCmethod.html
http://numericalmethods.eng.usf.edu/topics/shooting_method.htmlhttp://numericalmethods.eng.usf.edu/topics/shooting_method.htmlhttp://numericalmethods.eng.usf.edu/topics/shooting_method.htmlhttp://numericalmethods.eng.usf.edu/topics/shooting_method.html


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