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=
1
2
h
2
(u
00
1
+ u
00
0
[1 + hf
0
u
(x
1
, ˜y
1
)]) + O(h
3
) ,
v
j+1
=
1
2
h
2
j
X
k=0
u
00
k
j
Y
i=k+1
[1 + hf
0
u
(x
i
, ˜y
i
)] + O(h
3
) =
=
1
2
h
2
j
X
k=0
u
00
k
·
1 +
j
X
i=k+1
hf
0
u
(x
i
, ˜y
i
)
| {z }
≤c(x
j
−x
k+1
)
¸
| {z }
≤exp{c(x
j
−x
k+1
)}
+O(h
3
) .
u
00
= f
0
x
+ ff
0
u
|u
00
| ≤ c + cc ≡ c
1
|v
j+1
| ≤
1
2
hc
1
j
X
k=0
he
c(x
j
−x
k
)
=
1
2
hc
1
x
j
Z
x
0
e
c(x
j
−t)
dt + o(h) =
= h
c
1
2c
[e
c(x
j
−x
0
)
− 1] + o(h) = O(h) .
h
u(x)
u
j+1
= u
j
+ hf(x
j
, u
j
) +
h
2
2
[f
0
x
(x
j
, u
j
) + f(x
j
, u
j
)f
0
u
(x
j
, u
j
)]
| {z }
u
00
(x
j
)
+ . . . . (4)
f(x, u)
u
j+1
= u
j
+ h[αf(x
j
, u
j
) + βf (x
j
+ γh, u
j
+ δh)] + . . . , (5)
α, β, γ, δ
O(h
3
) f(x
j
+ γh, u
j
+ δh)
u
j+1
= u
j
+ h(α + β)f(x
j
, u
j
) + βh
2
[γf
0
x
(x
j
, u
j
) + δf
0
u
(x
j
, u
j
)] + O(h
3
) .
α + β = 1 βγ =
1
2
βδ =
1
2
f(x
j
, u
j
) β u
j
= u(x
j
) y
j
y
j+1
= y
j
+ h[(1 − β)f (x
j
, y
j
) + βf (x
j
+
h
2β
, y
j
+
h
2β
f(x
j
, y
j
))] , 0 < β ≤ 1 .
β
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