Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2
.pdf
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h0 |
δ |
h < h0 |
||ε(h)|| < δ |
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Lhw(h) = δf (h) + ε(h), |
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ε(h) |
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w(h) |
u(h) |
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w(h) − u(h) |
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||w(h) − u(h)|| C1||ε(h)||, |
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C1 |
h |
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u(h)
u |
hk |
u(h) |
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[u]h |
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|| u h − u(h)|| C · C1hk, |
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C C1 |
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f (h)
ε(h)
ε(h)
Lhw(h)
ε(h)
u(h)
h
Dh 
τ 
h
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(n+1,m) |
(n+1,m+1) |
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(n+1,m) |
(n+1,m+1) |
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(n+1,m) |
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τ |
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(n,m+1) |
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τ |
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(n,m-2) |
(n,m-1) |
(n,m) |
(n,m+1) |
(n,m+2) |
(n,m) |
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a, b, c |
u − (c/a)t = |
= f (x − (b/a)t) 
u − (c/a)t 
x − (b/a)t = const
(tn+1, xm) n + 1 
u −(c/a)t 
n + 1
(tn+1, xm) (n, m) 
(n, m + 1) 
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(n, m − 1) (n, m) |
h (b/a)τ |
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h < (b/a)τ |
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(n + 1, m) |
[(n, m − 1); (n, m)] |
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h < (b/a)τ |
h > (b/a)τ |
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τ /h 
b/a 
(n + 1, m) 
(n, m + 1)
unm 
unm + δunm 
unm 
a |
δumn+1 − δumn |
+ b |
δumn +1 − δumn |
= 0, |
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τ |
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h |
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a |
δumn+1 − δumn |
+ b |
δumn − δumn −1 |
= 0, |
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τ |
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h |
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x |
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a |
δumn+1 − δumn |
+ b |
δumn +1 − δumn −1 |
= 0, |
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τ |
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2h |
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a |
δumn+1 − δumn |
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+ b |
δumn+1+1 − δumn+1 |
= 0, |
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τ |
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h |
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δumn+1 − δumn |
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δumn+1 |
− |
n+1 |
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a |
+ b |
δum−1 |
= 0. |
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τ |
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h |
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δunm = λn · eiωmh,
i 
w 
λ
(n = 0) 
ω 
δu0m = λ0eiωmh = cos ωxm + i sin ωxm.
λ 
h → 0 τ → 0
|λn| N,
N
|λ| 1 + cτ,
c 
n |
| (1 + cτ ) |
T /τ |
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lim(1 + cτ )T /τ |
cT |
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N |
|λ |
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τ →0 |
= e |
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t = nτ = T |
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λ |
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ω |
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λ |
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λ = 0 |
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τ = rh r = const
λn · eiωmh |
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eiωh − 1 |
λ |
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a |
λ − 1 |
+ b |
= 0. |
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τ |
h |
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λ(ω) = 1 + ab (r − reiωh).
λ 
ω 
(b/a)r 
λ = 1 + (b/a)r |
r |
h 
τ
r = τ /h
a |
λ − 1 |
+ b |
1 − e−iωh |
= 0, |
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τ |
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h |
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λ = 1 − |
b |
(r + re−iωh). |
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a |
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λ |
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λ = 1 − |
ω |
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r |
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−(b/a)r |
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(b/a)r 1 τ (a/b)h |
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(b/a)r > 1 |
τ > (a/b)h
r = (τ /h) (a/b) 
r = (τ /h) > (a/b)
x
a λ − 1 + b eiωh − e−iωh = 0, τ 2h
λ 
τ /h
τ /h 
τ 
h
r
∂u∂t + (2et − x) ∂u∂x = 0, 
u(x, 0) = x
dt |
= |
dx |
= |
du |
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1 |
2et − x |
0 |
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