10/11/2015 1 Differentiation-Discrete Functions Chemical Engineering Majors Authors: Autar Kaw, Sri Harsha Garapati.
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04/19/23http://
numericalmethods.eng.usf.edu 1
Differentiation-Discrete Functions
Chemical Engineering Majors
Authors: Autar Kaw, Sri Harsha Garapati
http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM
Undergraduates
Differentiation –Discrete Functions
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Forward Difference Approximation
x
xfxxf
xxf
Δ
Δ
0Δ
lim
For a finite
'Δ' x
x
xfxxfxf
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x x+Δx
f(x)
Figure 1 Graphical Representation of forward difference approximation of first derivative.
Graphical Representation Of Forward Difference
Approximation
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Example 1
A new fuel for recreational boats being developed at the local university was tested at an area pond by a team of engineers. The interest is to document the environmental impact of the fuel – how quickly does the slick spread? Table 1 shows the video camera record of the radius of the wave generated by a drop of the fuel that fell into the pond.
Using the data
a)Compute the rate at which the radius of the drop was changing at seconds.
b)Estimate the rate at which the area of the contaminant was spreading across the pond at seconds.
2t
2t
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Example 1 Cont.
Use Forward Divided Difference approximation of the first derivative to solve the above problem. Use a time step of 0.5
sec.
Time 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Radius 0 0.236
0.667
1.225
1.886
2.635
3.464
4.365
5.333
Table 1 Radius as a function of time.
s t
m R
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Example 1 Cont.
Solution (a)
t
tRtRtR iii
1'
2it
5.0
25.2
1
ii ttt
5.21 it
5.0
25.22' RR
R
5.0
886.1635.2
m/s498.1
Example 1 Cont.
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(b) 2RArea
Time 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Area 0 0.17497 1.3977 4.7144 11.175 21.813 37.697 59.857 89.350
t
tAtAtA iii
1'
2it
5.025.2
1
ii ttt
5.21 it
s t
2m A
/sm276.215.0
175.11813.215.0
25.210
2
'
AAA
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Direct Fit Polynomials
'1' n nn yxyxyxyx ,,,,,,,, 221100 thn
nn
nnn xaxaxaaxP
1110
12121 12
)( n
nn
nn
n xnaxanxaadx
xdPxP
In this method, given data points
one can fit a order polynomial given by
To find the first derivative,
Similarly other derivatives can be found.
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Example 2-Direct Fit Polynomials
A new fuel for recreational boats being developed at the local university was tested at an area pond by a team of engineers. The interest is to document the environmental impact of the fuel – how quickly does the slick spread? Table 2 shows the video camera record of the radius of the wave generated by a drop of the fuel that fell into the pond. Using the data
(a)Compute the rate at which the radius of the drop was changing at seconds. (b)Estimate the rate at which the area of the contaminant was spreading across the pond at seconds.
Time (s) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Radius (m)
00.236
0.667 1.225 1.886 2.635 3.464 4.365 5.333
Use the third order polynomial interpolant for radius and area calculations.
Table 2 Radius as a function of time.
2t
2t
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Example 2-Direct Fit Polynomials cont.
33
2210 tatataatR
667.0,0.1 oo tRt
(a) For the third order polynomial (also called cubic interpolation), we choose the radius given by
Since we want to find the radius at , and we are using a third order polynomial, we need to choose the four points closest to and that also bracket to evaluate it.
The four points are
(Note: Choosing is equally valid.)
225.1,5.1 11 tRt
886.1,0.2 22 tRt
635.2,5.2 33 tRt
Solution
2t2t 2t
.5.2 and ,0.2 ,5.1 ,0.1 3210 tttt0.3 and ,5.2 ,0.2 ,5.1 3210 tttt
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Example 2-Direct Fit Polynomials cont.
33
2210 0.10.10.1667.00.1 aaaaR
33
2210 5.15.15.1225.15.1 aaaaR
33
2210 0.20.20.2886.10.2 aaaaR
33
2210 5.25.25.2635.25.2 aaaaR
635.2
886.1
225.1
667.0
625.1525.65.21
8421
375.325.25.11
1111
3
2
1
0
a
a
a
a
such that
Writing the four equations in matrix form, we have
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Example 2-Direct Fit Polynomials cont.
Solving the above four equations gives
0.0800000 a
0.471001 a
0.295992 a
0.0200003 a
33
2210 tatataatR
5.21,020000.029599.047100.0080000.0 32 tttt
Hence
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Example 2-Direct Fit Polynomials cont.
Figure 2 Graph of radius vs. time.
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Example 2-Direct Fit Polynomials cont.
5.21,020000.029599.047100.0080000.0 32 tttttR
tRdt
dtR '
32 020000.029599.047100.0080000.0 tttdt
d
5.21,060000.059180.047100.0 2 ttt
2' )2(0.060000-0.59180(2)47100.02 R
s/m 1.415
Given that
,
2
' 2
t
tRdt
dR
The derivative of radius at t=2 is given by
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Example 2-Direct Fit Polynomials cont.
Time 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Area 0 0.17497 1.3977 4.7144 11.175 21.813 37.697 59.857 89.350
2RArea
33
2210 tatataatA
(b)
For the third order polynomial (also called cubic interpolation), we choose thearea given by
Since we want to find the area at , and we are using a third order polynomial, we need to choose the four points closest to and that also bracket to evaluate it.
The four points are
(Note: Choosing is equally valid.) 3977.1,0.1 oo tAt
7144.4,5.1 11 tAt
175.11,0.2 22 tAt
813.21,5.2 33 tAt
2m A
s t
2t2t 2t
.5.2 and ,0.2 ,5.1 ,0.1 3210 tttt0.3 and ,5.2 ,0.2 ,5.1 3210 tttt
Example 2-fit Direct Ploynomials cont.
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33
2210 0.10.10.13977.10.1 aaaaA
33
2210 5.15.15.17144.45.1 aaaaA
33
2210 0.20.20.2175.110.2 aaaaA
33
2210 5.25.25.2813.215.2 aaaaA
813.21
175.11
7144.4
3977.1
625.1525.65.21
8421
375.325.25.11
1111
3
2
1
0
a
a
a
a
such that
Writing the four equations in matrix form, we have
Example 2- Direct Fit polynomials cont.
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0.0579000 a
0.120751 a0.0814682 a
1.37903 a
33
2210 tatataatA
5.21,3790.1081468.012075.0057900.0 32 tttt
Solving the above four equations gives
Hence
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Example 2-Direct Fit Polynomials cont.
Figure 3 Graph of area vs. time.
Example 2- Direct Fit Polynomial cont
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5.21,3790.1081468.012075.0057900.0 32 tttttA
tAdt
dtA '
32 3790.1081468.012075.0057900.0 tttdt
d
5.21,4.137116294.012075.0 2 ttt
2)2(4.13710.16294(2)12075.02 A
/sm 16.754 2
Given that
,
2
' 2
t
tEdt
dA
The derivative of radius at t=2 is given by
http://numericalmethods.eng.usf.edu21
Lagrange Polynomial nn yxyx ,,,, 11 thn 1In this method, given , one can fit a order Lagrangian polynomial
given by
n
iiin xfxLxf
0
)()()(
where ‘ n ’ in )(xf n stands for the thnorder polynomial that approximates the function
)(xfy given at )1( n data points as nnnn yxyxyxyx ,,,,......,,,, 111100 , and
n
ijj ji
ji xx
xxxL
0
)(
)(xLi a weighting function that includes a product of )1( n terms with terms of
ij omitted.
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Then to find the first derivative, one can differentiate xfnfor other derivatives.
For example, the second order Lagrange polynomial passing through
221100 ,,,,, yxyxyx is
2
1202
101
2101
200
2010
212 xf
xxxx
xxxxxf
xxxx
xxxxxf
xxxx
xxxxxf
Differentiating equation (2) gives
once, and so on
Lagrange Polynomial Cont.
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21202
12101
02010
2
222xf
xxxxxf
xxxxxf
xxxxxf
2
1202
101
2101
200
2010
212
222xf
xxxx
xxxxf
xxxx
xxxxf
xxxx
xxxxf
Differentiating again would give the second derivative as
Lagrange Polynomial Cont.
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Example 3
A new fuel for recreational boats being developed at the local university was tested at an are pond by a team of engineers. The interest is to document the environmental impact of the fuel – how quickly does the slick spread? Table 3 shows the video camera record of the radius of the wave generated by a drop of the fuel that fell into the pond. Using the data
(a)Compute the rate at which the radius of the drop was changing at .(b)Estimate the rate at which the area of the contaminant was spreading across the pond at .
Time 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Radius 0 0.236 0.667 1.225 1.886 2.635 3.464 4.365 5.333
Use second order Lagrangian polynomial interpolation to solve the problem.
Table 3 Radius as a function of time.
2t
2t
s t
m R
Since we want to find the radius at , and we are using a second order Lagrangian polynomial, we need to choose the three points closest to that also bracket to evaluate it. The three points are
Differentiating the above equation gives
http://numericalmethods.eng.usf.edu25
Example 3 Cont.
)()()()( 212
1
02
01
21
2
01
00
20
2
10
1 tRtt
tt
tt
tttR
tt
tt
tt
tttR
tt
tt
tt
tttR
2
1202
101
2101
200
2010
21' 222tR
tttt
ttttR
tttt
ttttR
tttt
ttttR
2.635
0.25.25.15.2
0.25.1221.886
5.20.25.10.2
5.25.1221.225
5.25.10.25.1
5.20.2222'
R
Solution:(a) For second order Lagrangian polynomial interpolation, we choose the radius given by
m/s 4100.1
. 5.2 and , 0.2 , 5.1 210 ttt
Hence,
2t2t 2t
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Example 3 Cont.
Time 0
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Area 0
0.17497
1.3977
4.7144
11.175
21.813
37.697
59.857
89.350
2RArea
)()()()( 212
1
02
01
21
2
01
00
20
2
10
1 tAtt
tt
tt
tttA
tt
tt
tt
tttA
tt
tt
tt
tttA
(b)
For second order Lagrangian polynomial interpolation, we choose the area given by
s t
m A
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Example 3 Cont.
4.7144
5.25.10.25.1
5.20.2222'
A
11.1755.20.25.10.2
5.25.122
21.8130.25.25.15.2
0.25.122
/sm 17.099 2
Since we want to find the area at , and we are using a second order Lagrangian polynomial, we need to choose the three points closest to that also brackets to evaluate it. The three points are
Differentiating the above equation gives
2t
2t2t
.5.2 and ,0.2 ,5.1 210 ttt
0
2010
21' 2tA
tttt
ttttA
12101
202tA
tttt
ttt
21202
102tA
tttt
ttt
Hence
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/discrete_02dif.html
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