Numerical Methods for Ordinary Dierential Equations Episode 3 pot
Numerical Methods for Ordinary Dierential Equations Episode 3 pot
... 0.00001416 −0.000 039 45
n y
3
error Ratio y
4
error Ratio
32 0.04 038 636 −0.01548159
3. 9497 4.1552
64 0.01022525 −0.0 037 2585
4.0 132 4.0659
128 0.002547 93 −0.00091 636
4.01 63 4.0294
256 0.000 634 40 −0.00022742
4.0105 ... y
2
error Ratio
32 0.01479021 −0.04016858
3. 9676 3. 9688
64 0.0 037 2781 −0.01012098
4.0417 4.0001
128 0.00092 233 −0.002 530 20
4. 036 1 4.0041
256 0.0002...
Numerical Methods for Ordinary Dierential Equations Episode 7 potx
... −
36 9
73
72
73
538 0
219
−
12285
584
2695
1752
8
9
−
8716
891
656
297
39 520
891
−
416
11
52
27
0
1
30 15
256
−
9
4
−
4219
78
5985
128
−
539
38 4
0
6 93
332 8
3
80
0
4
25
2 43
1120
77
160
73
700
00
57
640
0 −
16
65
137 7
2240
121
32 0
0
891
832 0
2
35
33
640
0 −
132
32 5
891
2240
−
33
32 0
−
73
700
891
832 0
2
35
.
As ... sides of (33 3m)– (33 3p) consist of onl...
Numerical Methods for Ordinary Dierential Equations Episode 10 pot
... b
4
=1,
b
2
c
2
+ b
3
c
3
+ b
4
c
4
=
1
2
,
b
2
c
2
2
+ b
3
c
2
3
+ b
4
c
2
4
=
1
3
,
b
3
a
32
c
2
+ b
4
a
42
c
2
+ b
4
a
43
c
3
=
1
6
,
b
2
c
3
2
+ b
3
c
3
3
+ b
4
c
3
4
=
1
4
,
b
3
a
32
c
2
(2c
3
− c
2
)+b
4
a
42
c
2
(2c
4
− ... r. The tableau for this h-adjusting step is
0
1
3
1
3
2
3
1
6
1
2
5
6
5
24
0
5
8
1
2
13
40
1
6
1
24
−
1
30
3+ r
3
−2r...
Numerical Methods for Ordinary Dierential Equations Episode 1 docx
... T
0
◦
6.2 831 8 530 72
3
◦
6.2842620 831
6
◦
6.2874944421
9
◦
6.2928884880
12
◦
6 .30 0454 431 1
15
◦
6 .31 02066 431
18
◦
6 .32 21 637 356
21
◦
6 .33 634 86 630
24
◦
6 .35 27888501
27
◦
6 .37 151 634 62
30
◦
6 .39 25680085
where q = ... Representing Methods in General Linear Form 37 3
500 Multivalue–multistage methods 37 3
501 Transformations of methods 37 5
502 Runge–Kutta...
Numerical Methods for Ordinary Dierential Equations Episode 2 docx
... y
2
n−1
, ( 133 a)
y
n
= y
2
n−1
− 2, ( 133 b)
y
n
= y
n−1
y
n−2
. ( 133 c)
If z
n
=ln(y
n
) in ( 133 a), then z
n
=2z
n−1
with solution z
n
=2
n
z
0
. Hence, the
general solution to ( 133 a) is
y
n
= ... equations of motion become
38 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
13 Difference Equation Problems
130 Introduction to difference equations
While differential...
Numerical Methods for Ordinary Dierential Equations Episode 4 doc
... ( 233 b)
b
2
c
2
2
+ b
3
c
2
3
=
1
3
, ( 233 c)
b
3
a
32
c
2
=
1
6
. ( 233 d)
The following tableaux
RK31 :
0
2
3
2
3
2
3
1
3
1
3
1
4
0
3
4
( 233 e)
and
RK32 :
0
1
2
1
2
1 −12
1
6
2
3
1
6
( 233 f)
give two possible ... conditions ( 233 a), ( 233 b), ( 233 c) and
( 233 d) such that b
1
=0,c
3
=1.
23. 3 Continue Table 234 (I) to include trees of order 5.
23. 4 Write...
Numerical Methods for Ordinary Dierential Equations Episode 5 docx
... 1]
1
2 3
[1, 2]
1
2
3
[1, 3]
1
3
2
[2, 1]
2
1
3
[2, 2]
2
1 3
[2, 3]
2
3
1
[3, 1]
3
1
2
[3, 2]
3
2
1
[3, 3]
3
1 2
30 3 The use of labelled trees
We have seen that α(t), introduced in Subsection 30 2, ... 7
8
−1
1.22674
3. 947 73
8
8
−2
3. 30401 × 10
−1
3. 71289
8
8
3
8.2 832 8 × 10
−2
3. 98876
10
8
−4
2 .33 986 × 10
−2
3. 54007
13
8
−5
4.95205 × 10
3
4.7...
Numerical Methods for Ordinary Dierential Equations Episode 6 ppsx
... c
3
c
3
c
3
(1−2c
3
)
2(1−c
3
)
c
3
2(1−c
3
)
1
12c
3
3
−24c
2
3
+1 7 c
3
−4
2(1−c
3
)(6c
3
−1−6c
2
3
)
c
3
(1−2c
3
)
2(1−c
3
)(6c
3
−1−6c
2
3
)
1−c
3
6c
3
−1−6c
2
3
6c
3
−1−6c
2
3
12c
3
(1−c
3
)
1
12c
3
(1−c
3
)
1
12c
3
(1−c
3
)
6c
3
−1−6c
2
3
12c
3
(1−c
3
)
;
(32 2e)
for ... defining equations are
used in the choice of pa...
Numerical Methods for Ordinary Dierential Equations Episode 8 ppsx
... =5,p=8),
0
1
20
−7−
√
21
120
1
15
−7+
√
21
120
0
7−
√
21
14
1
20
34 3+9
√
21
2520
56−15
√
21
31 5
34 3−69
√
21
2520
0
1
2
1
20
49+12
√
12
36 0
8
45
49−12
√
12
36 0
0
7+
√
21
14
1
20
34 3+69
√
21
2520
56+15
√
21
31 5
34 3−9
√
21
2520
0
1
1
20
119 3
√
21
36 0
13
45
119 +3
√
21
36 0
0
1
20
49
180
16
45
49
180
1
20
Lobatto ... (35 7f) and (35 7g) and rearrange to deduce (35 7e).
...
Numerical Methods for Ordinary Dierential Equations Episode 9 ppsx
... +
3
2
ξ
2
−
1
6
ξ
3
41− 4ξ +3
2
−
2
3
ξ
3
+
1
24
ξ
4
51− 5ξ +5ξ
2
−
5
3
ξ
3
+
5
24
ξ
4
−
1
120
ξ
5
61− 6ξ +
15
2
ξ
2
−
10
3
ξ
3
+
5
8
ξ
4
−
1
20
ξ
5
+
1
720
ξ
6
71− 7ξ +
21
2
ξ
2
−
35
6
ξ
3
+
35
24
ξ
4
−
7
40
ξ
5
+
7
720
ξ
6
−
1
5040
ξ
7
81− ... for
296 NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
Table 38 6(III) Formulae for (α
−1
)(t
i
) up to...
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