7/2/2015 1 Differentiation-Discrete Functions Industrial Engineering Majors Authors: Autar Kaw, Sri...
Transcript of 7/2/2015 1 Differentiation-Discrete Functions Industrial Engineering Majors Authors: Autar Kaw, Sri...
04/19/23http://
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Differentiation-Discrete Functions
Industrial 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
The failure rate of a direct methanol fuel cell (DMFC) is given by the formula
Where is the reliability at a certain time , and the values of the reliability are given in Table 1.
th
tR
tRtR
th'
t
0 1 10 100 1000 2000 3000 4000 5000
1 0.9999 0.9998 0.9980 0.9802 0.9609 0.9419 0.9233 0.9050 hrs t
Table 1 Reliability of DMFC system.
tR
Using the forward divided difference method, find the failure rate of the DMFC system at hours.50t
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Example 1 Cont.
t
tRtRtR iii
1'
10it
9010100
1
ii ttt
5100000.290
9998.09980.090
1010050'
RRR
1001 it
Solution
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Example 1 Cont.
999.0
9998.040100000.2
10105010100
1010050
5
RRR
R
5
5
100020.2
999.0
100000.2
50
50'50
R
Rh
The reliability at hours is tR 50t
The failure rate at hours is then th 50t
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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
The failure rate of a direct methanol fuel cell (DMFC) is given by the formula
Where is the reliability at a certain time , and the values of the reliability are given in Table 2.
th
tR
tRtR
th'
t
0 1 10 100 1000 2000 3000 4000 5000
1 0.9999 0.9998 0.9980 0.9802 0.9609 0.9419 0.9233 0.9050 hrs t
Table 2 Reliability of DMFC system.
tR
Using a third order polynomial interpolant for reliability , find the failure rate of the DMFC system at hours.50t
tR
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Example 2-Direct Fit Polynomials cont.
For the third order polynomial (also called cubic interpolation), we choose the reliability given by 3
32
210 tatataatR Since we want to find the reliability at , and we are using third order polynomial, we need to choose the four points closest to and that also bracket to evaluate it.
The four points are , , and hours.
9802.0,1000
9980.0,100
9998.0,10
9999.0,1
33
22
11
tRt
tRt
tRt
tRt oo
50t50t 50t
10 t 101 t 1002 t 10003 t
Solution
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Example 2-Direct Fit Polynomials cont.
such that
Writing the four equations in matrix form, we have
3
32
210
33
2210
33
2210
33
2210
1000100010009802.01000
1001001009980.0100
1001109998.010
1119999.01
aaaaR
aaaaR
aaaaR
aaaaR
9802.0
9980.0
9998.0
9999.0
10110110001
101100001001
1000100101
1111
3
2
1
0
96
6
a
a
a
a
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Example 2-Direct Fit Polynomials cont.
Solving the above four equations gives
113
82
51
0
100101.9
109788.9
100023.1
9991.0
a
a
a
a
Hence
10001 ,100101.9109788.9100023.199991.0 311285
33
2210
tttt
tatataatR
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Example 2-Direct Fit Polynomials cont.
Figure 2 Graph of reliability as a function of time.
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Example 2-Direct Fit Polynomials cont.
The reliability at is given by,
50
50'
t
tRdt
dR
Given that
10001 ,100101.9109788.9100023.199991.0 311285 tttttR
,
10001 ,107030.2109958.1100023.1
100101.9109788.9100023.199991.0
'
21075
311285
ttt
tttdt
d
tRdt
dtR
50t
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Example 2-Direct Fit Polynomials cont.
5
21075
109326.1
50107030.250109958.1100023.150'
R
Using the same function, we can also calculate the value of at .
tR 50t
99917.0
50100101.950109788.950100023.199991.050
10001 ,100101.9109788.9100023.199991.0311285
311285
R
tttttR
The failure rate is then
5
5
109343.1
99917.0
109326.1
'
tR
tRth
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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
The failure rate of a direct methanol fuel cell (DMFC) is given by the formula
Where is the reliability at a certain time , and the values of the reliability are given in Table 3.
th
tR
tRtR
th'
t
0 1 10 100 1000 2000 3000 4000 5000
1 0.9999 0.9998 0.9980 0.9802 0.9609 0.9419 0.9233 0.9050 hrs t
Table 3 Reliability of DMFC system.
tR
Determine the value of the failure rate at hours using the second order Lagrangian polynomial interpolation for reliability.
50t
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Solution
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
21 222' tR
tttt
ttttR
tttt
ttttR
tttt
ttttR
For second order Lagrangian polynomial interpolation, we choose the reliability given by
Since we want to find the reliability 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 , , and .
Differentiation the above equation gives.
50t
50t 50t10 t 101 t 1002 t
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Hence
Example 3 Cont.
99918.0
9980.010100
1050
1100
150
9998.010010
10050
110
1509999.0
1001
10050
101
105050
R
5109102.1
9980.0101001100
1015029998.0
10010110
10015029999.0
1001101
1001050250'
R
We must also find the value of at . tR 50t
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Example 3 Cont.
5
5
109118.1
99918.0
109102.1
'
tR
tRth
The failure rate is then
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