shooting method
DESCRIPTION
Shooting Method. Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Shooting Method http://numericalmethods.eng.usf.edu. Shooting Method. - PowerPoint PPT PresentationTRANSCRIPT
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04/19/23http://
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Shooting Method
Major: All Engineering Majors
Authors: Autar Kaw, Charlie Barker
http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM
Undergraduates
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Shooting Method
http://numericalmethods.eng.usf.edu
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Shooting Method
The shooting method uses the methods used in solving initial value problems. This is done by assuming initial values that would have been given if the ordinary differential equation were a initial value problem. The boundary value obtained is compared with the actual boundary value. Using trial and error or some scientific approach, one tries to get as close to the boundary value as possible.
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Example
ba
r
0030770.08
,0038731.05
,01
22
2
u
ur
u
dr
du
rdr
ud
Let
wdr
du
Then
01
2
r
uwrdr
dw
Where a = 5and b = 8
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Solution
Two first order differential equations are given as
0038371.05, uwdr
du
knownnotwr
u
r
w
dr
dw 5,
2
Let us assume
00026538.0
58
5855
uu
dr
duw
To set up initial value problem
0038371.05,,,1 uwurfwdr
du
00026538.05,,,22 wwurf
r
u
r
w
dr
dw
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Solution Cont
Using Euler’s method,
hwurfuu iiiii ,,11
hwurfww iiiii ,,21
Let us consider 4 segments between the two boundaries, and then,
5r8r
75.04
58
h
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Solution Cont
00026538.0,0038371.0,5,0 000 wuriFor
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 Cont
For ,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 Cont
For 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 Cont25.775.050.6,3 23 hrri 000053332.0,0035832.0 33 wuFor
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
0036232.08 4 uu
So at
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Solution Cont
Let us assume a new value for 5dr
du
00053076.000026538.0258
582525
uu
dr
duw
Using 75.0h and Euler’s method, we get
"0029665.08 4 uu
While the given value of this boundary condition is
0030770.08 4 uu
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Solution Cont
Using linear interpolation on the obtained data for the two assumed values of
5dr
du we get
00030770.08 u
00026538.00036232.00030770.00036232.00029645.0
00026538.000053076.05
dr
du
00048611.0
Using 75.0h and repeating the Euler’s method with 00048611.0)5( w
0030769.08 4 uu
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Solution Cont
Using linear interpolation to refine the value of 4u
till one gets close to the actual value of 8u which gives you,
0038731.051 uu
0035085.075.5 2 uu
0032858.050.6 3 uu
0031518.025.7 4 uu
0030770.000.8 5 uu
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Comparisons of different initial guesses
3.0E-03
3.2E-03
3.4E-03
3.6E-03
3.8E-03
4.0E-03
5 6 7 8Radial Location, r (in)
Rad
ial
Dis
pla
cem
ent,
u
(in
)
du/dr = -0.00026538
du/dr = -0.00053076
du/d r= -0.00048611
Exact
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Comparison of Euler and Runge-Kutta Results with
exact results
Table 1 Comparison of Euler and Runge-Kutta results with exact results.
r (in) Exact (in) Euler (in)Runge-
Kutta (in)
55.756.57.25
8
3.8731×10−3
3.5567×10−3
3.3366×10−3
3.1829×10−3
3.0770×10−3
3.8731×10−3
3.5085×10−3
3.2858×10−3
3.1518×10−3
3.0770×10−3
0.00001.37311.5482
9.8967×10−1
1.9500×10−3
3.8731×10−3
3.5554×10−3
3.3341×10−3
3.1792×10−3
3.0723×10−3
0.00003.5824×10−
2
7.4037×10−
2
1.1612×10−
1
1.5168×10−
1
%t %t
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Additional ResourcesFor all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit
http://numericalmethods.eng.usf.edu/topics/shooting_method.html