Re: [微積] 偏微分
※ 引述《iverson32 (iverson32)》之銘言：
: 1 1
: (D.E.) Urr+-Ur+ - Uoo=0 Uoo代表對θ偏微兩次
: r r^2
: a<=r<b,-∞<=θ<=∞
: (B.C.) U(a,θ)=U(b,θ)=0
: -∞<=θ<=∞
R = 0 ,
Θ'' - λΘ = 0
2
r R'' + r R' + λ R = 0
2
λ < 0 , λ = -ω ,
ω -ω
R = c1 r + c2 r
ω -ω
R(a) = c1 a + c2 a = 0
ω -ω
R(b) = c1 b + c2 b = 0 , 因為 det ≠ 0 , c1 , c2 = 0
R(r) = 0 , trivial sol.
λ = 0 ,
R(r) = c1 + c2 lnr
R(a) = c1 + c2 lna = 0
R(b) = c1 + c2 lnb = 0 , 因為 det ≠ 0 , c1 , c2 = 0
R(r) = 0 , trivial sol.
2
λ > 0 , λ = ω
R(r) = c1 cos(ωlnr) + c2 sin(ωlnr)
R(a) = c1 cos(ωlna) + c2 sin(ωlna) = 0
R(b) = c1 cos(ωlnb) + c2 sin(ωlnb) = 0
det = 0 ,
cos(ωlna)sin(ωlnb) - sin(ωlna)cos(ωlnb) = 0
sin(ω[lnb - lna]) = 0
nπ
ω = ───── , n = 1 , 2 , 3 ....
lnb - lna
ωθ -ωθ
Θ = c1 e + c2 e
U = R(r)Θ(θ)
∞ ωθ -ωθ
= Σ {An e + Bn e } sin(ωr )
n=1
θ(-∞) = 有界
θ(∞) = 有界
....
U = 0
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有錯請不吝指正 謝謝 :)
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01/03 01:36, , 1F
01/03 01:36, 1F
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