Численные методы. Ч.2. Решение уравнений. Буслов В.А - 37 стр.

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СПбГУ | Санкт-Петербург

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N F = (F
1
, F
2
. . . , F
N
)
T
e
h
d
dx
F (x) =
F (x + h) e
±h
d
dx
F
i
= F
i±1
(
[e
−h
d
dx
+ 2 − e
h
d
dx
]F
i
= µF
i
,
F
0
= F
N+1
= 0.
F
(
[e
−h
d
dx
+ 2 − e
h
d
dx
]F = µF,
F
0
= F
N+1
= 0.
d
dx
1
i
d
dx
¿
1
i
d
dx
f, g
À
=
1
i
Z
f
0
(x)¯g(x)dx =
Z
f(x)
µ
1
i
d
dx
g(x)
dx =
¿
f,
1
i
d
dx
g
À
.
D e
ipx
1
i
d
dx
e
ipx
= pe
ipx
p ∈ R
1
A
f(A) f(A)
f(p) p A
Aϕ =
P
λ
i
hϕ, F
(i)
iF
(i)
f(A)ϕ =
P
f(λ
i
)hϕ, F
(i)
iF
(i)
⇒ f(A)F
(k)
= f(λ
k
)F
(k)
.
F = e
ipx
D f(D) =
[−e
ihD
− e
−ihD
+ 2]
[−e
ihD
− e
−ihD
+ 2]F = [−e
iph
− e
−iph
+ 2]F = 2[1 − cos ph]F .
f(D) f(p) = f(−p) e
−ipx
2[1 −cos(ph)] e
ipx
F (0) = F (a) = 0
F
0
= 0 p
F
p
j
= sin px
j
x
j
= hj F
N+1
= 0
sin ph(N + 1) = 0 ph(N + 1) = πn p
n
=
πn
h(N+1)
=
π n
b
n = 1, 2, . . . , N
F
n
: F
n
j
= sin
πn
b
x
j
, x
j
= hj .
Φ
n
x
j
j n
Φ
n
(x
j
) = F
n
j
.
λ
n
˜
λ
n
=
µ
n
h
2
˜
λ
n
=
µ
n
h
2
=
2
h
2
[1 − cos p
n
h] =
2
h
2
[1 − cos
πn
b
h] =
=
2
h
2
[1 − 1 +
π
2
n
2
2b
2
h
2
+ O(h
4
)] =
π
2
n
2
b
2
+ O(h
2
) = λ
n
+ O(h
2
) .

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