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y“ꊩÛ
1lÙ ~‡©•§êŠ){
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1lÙ ~‡©•§êŠ){
8.5 ˜‡©•§|Úp‡©•§
∗
c¡0˜~‡©•§ˆ«êŠ•{, ù•{é~‡©
•§|Úp~‡©•§Ó·^. •;•ÖþE,, e¡±
ü‡™•¼ê•§|Ú~‡©•§•~5Qãù•{O
Žúª, ÙäØÚíL§†˜•§œ/˜, Ø2K
ã, •ÑOŽ‚ª.
8.5.1 ˜~‡©•§|
•Ä•§|
y
0
= f(x, y, z), y(x
0
) = y
0
,
z
0
= g(x, y, z), z(x
0
) = z
0
.
(8.32)
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1. î.úª
é n = 0, 1, 2, · · · , OŽ
y
n+1
= y
n
+ hf(x
n
, y
n
, z
n
), y(x
0
) = y
0
,
z
n+1
= z
n
+ hg(x
n
, y
n
, z
n
), z(x
0
) = z
0
.
(8.33)
2. U?î.úª
é n = 0, 1, 2, · · · , OŽ
p
n+1
= y
n
+ hf(x
n
, y
n
, z
n
),
q
n+1
= z
n
+ hg(x
n
, y
n
, z
n
),
y
n+1
= y
n
+
h
2
f(x
n
, y
n
, z
n
) + f(x
n+1
, p
n+1
, q
n+1
)
,
z
n+1
= z
n
+
h
2
g(x
n
, y
n
, z
n
) + g(x
n+1
, p
n+1
, q
n+1
)
,
(8.34)
Ù¥ y(x
0
) = y
0
, z(x
0
) = z
0
.
3. ²;o9‚–¥©úª
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é n = 0, 1, 2, · · · , OŽ
y
n+1
= y
n
+
h
6
K
1
+ 2K
2
+ 2K
3
+ K
4
,
z
n+1
= z
n
+
h
6
L
1
+ 2L
2
+ 2L
3
+ L
4
,
(8.35)
Ù¥
K
1
= f(x
n
, y
n
, z
k
), L
1
= g(x
n
, y
n
, z
n
),
K
2
= f(x
n
+
h
2
, y
n
+
hK
1
2
, z
n
+
hL
1
2
), L
2
= g(x
n
+
h
2
, y
n
+
hK
1
2
, z
n
+
hL
1
2
),
K
3
= f(x
n
+
h
2
, y
n
+
hK
2
2
, z
n
+
hL
2
2
), L
3
= g(x
n
+
h
2
, y
n
+
hK
2
2
, z
n
+
hL
2
2
),
K
4
= f(x
n
+ h, y
n
+ hK
3
, z
n
+ hL
3
), L
4
= g(x
n
+ h, y
n
+ hK
3
, z
n
+ hL
3
).
4. oæd킪
P f
n−k
, g
n−k
©OL« f(x
n−k
, y
n−k
, z
n−k
), g(x
n−k
, y
n−k
, z
n−k
) (k =
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0, 1, 2, 3). é n = 0, 1, 2, · · · , OŽ
y
n+1
= y
n
+
h
24
55f
n
− 59f
n−1
+ 37f
n−2
− 9f
n−3
,
z
n+1
= z
n
+
h
24
55g
n
− 59g
n−1
+ 37g
n−2
− 9g
n−3
,
(8.36)
Ù¥ y(x
0
) = y
0
, z(x
0
) = z
0
.
5. oædý-‚ª
P f
n−k
, g
n−k
©OL« f(x
n−k
, y
n−k
, z
n−k
), g(x
n−k
, y
n−k
, z
n−k
) (k =
0, 1, 2, 3). é n = 0, 1, 2, · · · , OŽ
p
n+1
= y
n
+
h
24
55f
n
− 59f
n−1
+ 37f
n−2
− 9f
n−3
,
q
n+1
= z
n
+
h
24
55g
n
− 59g
n−1
+ 37g
n−2
− 9g
n−3
,
y
n+1
= y
n
+
h
24
f
n−2
− 5f
n−1
+ 19f
n
+ 9f(x
n+1
, p
n+1
, q
n+1
)
,
z
n+1
= z
n
+
h
24
g
n−2
− 5g
n−1
+ 19g
n
+ 9g(x
n+1
, p
n+1
, q
n+1
)
,
(8.37)
