Антоник В.Г. Численные методы: математический анализ и дифференциальные уравнения
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y
0
= y , y(0) = 1 , x ∈ [0, 4] ,
h = 1
y
0
= cos x + y , y(0) = 0 , x ∈ [0, 2π] ,
h =
π
2
[a, b]
y
0
i
= ha
i
(x), yi + f
i
(x) , i = 1, n .
y = y(x) n
a
i
(x), i = 1, n n f
i
(x)
i = 1, n
x
1
= a, x
2
= b
½
hb
i
, y(x
1
)i = β
i
, i = 1, k ,
hb
i
, y(x
2
)i = β
i
, i = k + 1, n ,
b
i
∈ R
n
β
i
y(x)
z ∈ [a, b]
H(c, x, y) =
n
X
i=1
c
i
³
ha
i
(x), yi + f
i
(x)
´
n
c = c(x)
c
0
i
= −
∂H
∂y
i
, i = 1, n ,
α
0
= c
1
f
1
(x) + . . . + c
n
f
n
(x) ,
α = α(x)
c
i
(x), α
i
(x)
½
c(x
1
) = b
i
,
α(x
1
) = β
i
, i = 1, k ,
(4.1)
½
c(x
2
) = b
i
,
α(x
2
) = β
i
, i = k + 1, n .
(4.2)
z x
1
z = x
2
z 6= x
1
, z 6=
x
2
z
hc
i
(z), y(z)i = α
i
(z) , i = 1, n .
y(z) y = y(z)
y
0
i
= ha
i
(x), yi + f
i
(x) , y
i
(z) = y
i
, i = 1, n .
y = y(x)
½
y
0
1
= 2x ,
y
0
2
= 3y
1
,
y
1
(0) + y
2
(0) = 1 ,
−2y
1
(2) + y
2
(2) = 4 .
z = 0
n = 2
x
1
= 0, x
2
= 2
x
1
k = 1
x
2
n − k = 1
b
1
= (1 1)
T
, β
1
= 1, b
2
= (−2 1)
T
, β
2
= 4
f
i
(x), i = 1, 2
f
1
(x) = 2x , f
2
(x) = 0
H(c, y, x) = 2xc
1
+ 3 y
1
c
2
.
c
0
1
= −
∂H
∂y
1
= −3c
2
,
c
0
2
= −
∂H
∂y
2
= 0 ,
α
0
= 2xc
1
.
z = 0 = x
1
c
1
(2) = −2 ,
c
2
(2) = 1 ,
α(2) = 4 .
c
1
(x) = 4 − 3x , c
2
(x) = 1 , α(x) = 4 + 4x
2
− 2x
3
.
z = 0
c
1
(0)y
1
(0) + c
2
(0)y
2
(0) = α(0) .
4y
1
(0) + y
2
(0) = 4 .
z = 0
y
1
(0) + y
2
(0) = 1 .
½
4y
1
(0) + y
2
(0) = 4 ,
y
1
(0) + y
2
(0) = 1 .
y(0) =
µ
1
0
¶
.
½
y
0
1
= 2x ,
y
0
2
= 3y
1
,
y
1
(0) = 1 ,
y
2
(0) = 0 .
y
1
(x) = x
2
+ 1 , y
2
(x) = x
3
+ 3 x . ¤
z
½
y
0
1
= 3y
2
+ 2 ,
y
0
2
= 2x ,
y
1
(0) − y
2
(0) = 1 ,
y
1
(2) − 3y
2
(2) = 1 , z = 1 .
½
y
0
1
= −4x ,
y
0
2
= 6y
1
− x ,
y
1
(0) − y
2
(0) = 0 ,
y
1
(1) + 2y
2
(1) = 4 , z = 0 .
(
y
0
1
= y
2
,
y
0
2
= 2 − y
1
,
y
1
(0) + y
2
(0) = 1 ,
2y
1
³
π
2
´
− y
2
³
π
2
´
= 0 , z = 0 .
½
y
0
1
= y
2
− 2x ,
y
0
2
= 4 ,
y
1
(0) + y
2
(0) = −2 ,
2y
1
(1) + y
2
(1) = 3 , z = 1 .
y
0
1
= 2 ,
y
0
2
= 3y
3
+ y
1
,
y
0
3
= y
1
,
y
1
(0) − y
2
(0) = −2 ,
y
2
(1) + y
3
(1) = 1 ,
y
1
(1) − y
3
(1) = 2 , z = 1 .
[a, b]
y
00
− p(x)y = f (x) ,
y(a) = y
a
, y(b) = y
b
.
y = y(x) p(x)
f(x) y
a
, y
b
g
0
(x), g
1
(x), . . . ,
g
m
(x), x ∈ [a, b]
g
i
(x) [a, b]
g
0
(a) = y
a
, g
0
(b) = y
b
, g
i
(a) = 0, g
i
(b) = 0, i = 1, m
g
i
(x), i = 1, m
y(x, α) = g
0
(x) + a
1
g
1
(x) + . . . + a
m
g
m
(x) ,
α = (a
1
, a
2
, . . . , a
m
)
ϕ(x, α) = y
00
(x, α) − p(x)y(x, α) − f (x) .
ϕ(x, α)
y(x, α)
y(x)
a
1
, a
2
, . . . , a
m
ϕ(x, α) ≡ 0 , a ≤ x ≤ b ,
y(x, α) y(x, α) = y(x)
y(x, α)
(a, b) m x
1
, . . . , x
m
a < x
1
<
. . . < x
m
< b
ϕ
ϕ(x
i
, α) = 0 , i = 1, m .
a
1
, . . . , a
m
α
∗
= (a
∗
1
, . . . , a
∗
m
)
˜y(x) = y(x, α
∗
) = g
0
(x) + a
∗
1
g
1
(x) + . . . + a
∗
m
g
m
(x) .
s(x), a ≤ x ≤ b
Ls(x)
Ls(x)
4
= s
00
(x) − p(x)s(x) .
a
1
Lg
1
(x
1
) + . . . + a
m
Lg
m
(x
1
) = f(x
1
) − Lg
0
(x
1
) ,
a
1
Lg
1
(x
2
) + . . . + a
m
Lg
m
(x
2
) = f(x
2
) − Lg
0
(x
2
) ,
. . .
a
1
Lg
1
(x
m
) + . . . + a
m
Lg
m
(x
m
) = f(x
m
) − Lg
0
(x
m
) .
g
i
(x), i = 1, n
g
i
= (x − a)
i
(x − b)
g
i
= (x − a)(x − b)
i
g
i
= sin
³
iπ
x − a
b − a
´
g
0
(x)
y
a
= y
b
g
0
(x) = y
a
y
a
6= y
b
g
0
(x) =
1
b − a
³
(x − a)y
b
− (x − b)y
a
´
.
y
00
− y = 1 , 0 ≤ x ≤ 3 ,
y(0) = 0 , y(3) = 0 .
m = 2
a = 0 , b = 3 , p(x) = 1 , f(x) = 1 , y
a
= 0 , y
b
= 0 .