Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2
.pdfu(r, ϕ)
u(r, ϕ)|r=R = ψ(R cos ϕ, R sin ϕ) = f (ϕ),
f (ϕ) L r = R
u(r, ϕ) = R(r)(ϕ).
(ϕ)R (r) + 1r (ϕ)R (r) + r12 R(r) (ϕ) = 0.
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rR (r) + r2R (r) |
= |
(ϕ) |
= C. |
R(r) |
(ϕ) |
C
C = −λ2.
(ϕ) + λ2 (ϕ) = 0,
r2R (r) + rR (r) − λ2R(r) = 0.
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(ϕ) = A cos(λϕ) + B sin(λϕ). |
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ϕ |
2π |
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(r, ϕ) |
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ϕ |
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2π |
λ = 1, 2, 3, . . . |
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k(ϕ) |
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k(ϕ) = Ak cos(kϕ) + Bk sin(kϕ), k = 1, 2, 3, . . . |
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λ |
λ |
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Bk |
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R(r) = rm |
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R (r) = mrm−1, R (r) = m(m |
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1)rm−2 |
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r2m(m − 1)rm−2 + rmrm−1 − λ2rm = 0 |
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2 |
2 |
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m |
− λ = 0 |
m1,2 = ±λ |
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R1(r) = rλ |
R2(r) = r−λ |
R2(r) = r−λ |
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r → 0 |
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R(r) = Crλ λ = k
Rk(r) = Ckrk.
uk(r, ϕ)
uk(r, ϕ) = Rk(r)k(ϕ) = (Ak cos(kϕ) + Bk sin(kϕ))Ckrk = = (ak cos(kϕ) + bk sin(kϕ))rk, k = 1, 2, 3, . . .
u0(r, ϕ) = a0/2
u(r, ϕ)
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∞ |
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a0 |
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(ak cos(kϕ) + bk sin(kϕ))rk. |
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u(r, ϕ) = |
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2 |
k=1 |
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ak |
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bk |
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a0 |
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u(r, ϕ)|r=R = |
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2 |
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(akRk cos(kϕ) + bkRk sin(kϕ)) = f (ϕ), |
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k=1 |
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π |
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π |
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1 |
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1 |
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a0 = |
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f (ϕ)dϕ, |
akRk = |
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f (ϕ) cos(kϕ)dϕ, |
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π |
π |
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−π |
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−π |
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π |
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bkRk = |
1 |
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f (ϕ) sin(kϕ)dϕ, |
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k = 1, 2, 3, . . . |
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−π |
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a0 = |
π |
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f (ϕ)dϕ, |
ak = |
Rkπ |
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−π |
f (ϕ) cos(kϕ)dϕ, |
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−π |
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bk = |
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−π f (ϕ) sin(kϕ)dϕ, |
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k = 1, 2, 3, . . . |
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Rkπ |
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ak bk
R
10C 00C
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f (ϕ) = |
1, |
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0 ϕ π |
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0, |
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π < ϕ < 2π. |
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1 |
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2π |
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a0 = |
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1dϕ + |
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ak = |
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1 cos(kϕ)dϕ = 0. |
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π |
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π |
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0 |
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π |
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0dϕ = 1, |
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πRk |
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b |
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= |
1 |
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0 |
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1 sin(kϕ)dϕ = |
1 − (−1)k |
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b |
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= 0, b |
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= |
2 |
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k |
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2n |
2n−1 |
πR2n−1(2n − 1) |
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πRk |
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πRkk |
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u(r, ϕ) = |
1 + 2 |
∞ |
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r |
2n−1 sin((2n − 1)ϕ) . |
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2 π n=1 |
R |
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2n − 1 |
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1 |
2π |
R2 − τ 2 |
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u(r, ϕ) = |
f (τ ) |
dτ. |
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2π |
0 |
R2 − 2Rτ cos(τ − ϕ) + τ 2 |
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0z
a b
M x N y x y hx hy
y
b |
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m,n |
0 |
x |
a |
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∂2u |
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∂2u |
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(m, n) |
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∂x2 |
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∂y2 |
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um+1,n − 2um,n + um−1,n + um,n+1 − 2um,n + um,n−1 = 0, |
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− |
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− |
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hx2 |
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hy2 |
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n = 1, 2, . . . N |
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1; m = 1, 2, . . . , M |
1, |
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u |
m,0 |
= ϕ(x |
m |
, 0), |
m = 0, 1, . . . , M, |
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m = 0, 1, . . . , M, |
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um,N = ϕ(xm, b), |
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= ϕ(0, yn), |
n = 1, 2, . . . , N − 1, |
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u0,n |
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n = 1, 2, . . . , N 1. |
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uM,n = ϕ(a, yn), |
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x y
U
(N + 1)(M + 1)
N M
um,n
(p + 1) u (p)
um,n(p+1) = um(p+1) |
,n + um(p−) |
1,n + r um,n(p) |
+1 + um,n(p) −1 |
/ (2(r + 1)) , |
r = (hx/hy)2 u
um,n
ε
|u(m,np+1) − u(m,np) | < ε.
um,n(p+1) = |
ω |
(um(p+1) |
,n + um(p−) |
1,n + r(um,n(p) |
+1 + um,n(p) |
−1)) + (1 −ω)um,n(p) , |
2(r + 1) |
ω
ω
ω
u(0)
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D : |
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∂u |
= A |
∂2u |
+ |
∂2u |
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∂t |
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∂x2 |
∂y2 |
(t, x, y)
tn = nτ, |
n = 0, 1, 2, . . . |
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xm = mh, |
m = 0, ±1, ±2, . . . |
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y |
k |
n |
n |
± |
± |
2, . . . |
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= kl, |
k = 0, |
1, |
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u(t , xm, yk) = Um,k.
δUm,kn = λn · etw1mh · etw2kl.
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∂u |
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πx |
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u |
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= sin πx, |
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= 0, 5 · cos 2 |
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t=0 |
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∂t t=0 |
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= 0, |
u + |
∂u |
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u x=0 |
∂x x=L = 0, 1 sin 2πt. |
um0 |
= sin πxm, |
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m = 0, 1, 2, ..., M ; |
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um1 |
− um0 |
= 0, 5 |
· |
cos |
πxm |
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m = 0, 1, ..., M ; |
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2 |
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τ |
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un+1 |
= 0, |
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= 0, 1, 2, ..., N |
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1; |
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0 |
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n+1 |
n n+1 |
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n+1 |
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uM |
− uM−1 |
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n+1 |
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uM |
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+ |
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h |
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= 0, 1 · sin 2πt |
, n = 0, 1, ..., N − 1. |
u0m = sin πxm.
u1m = u0m + τ · cos(0, 5πxm).
un0 +1 = 0.
unM+1 = unM+1−1 + h · 0.1 · sin(2πtn+1) /(1 + h).
h
0, 05 τ r · h
r 1 r
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B2 |
D2 |
F 2 |
r h |
τ |
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= B2 D2 |
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A3 |
t/x |
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xm |
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B3 |
C3 |
= D2 |
D3 |
= C3+$D$2 |
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V 3 x = 1 |
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t |
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A4
A5 = $F $2 A6 = A5 + $F $2
A24 |
B4 : V 4 |
= sin(3, 14159 B$3) |
B4 |
L4 |
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= C4 + $F $2 cos(1, 5708 C$3) |
B5 |
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C6 |
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U 6 |
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B6 |
= 0 |
V 5 |
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= (U 6 + $D$2 0, 1sin(6, 28 $A5)/(1 + $D$2)) A6 : V 6
t = 0; 0.2; 0.4; 0.6; 0.8 1.0