Differential Equations Lecture Work Solutions 120

Differential Equations Lecture Work Solutions 120 - 2. u...

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Unformatted text preview: 2. u (r, θ, 0) = 0 = uz (r, θ, H ) u (r, 0, z ) = 0 u (a, θ, z ) = u (r, π, z ) = β (θ, z ) u = R(r ) Θ (θ) Z (z ) r (r R ) + (λ r 2 − µ) R = 0 Z + λZ = 0 Θ + µΘ = 0 Z (0) = 0 Θ (0) = Θ (π ) = 0 | R (0) | < ∞ ⇓ Z (H ) = 0 ⇓ Zn = sin n − 1 2 π H z r (r R ) − Θm = sin m θ n− 1 2 π H 2 r 2 + m2 µm = m2 λn = 2 π H 1 2 n− m = 1, 2, · · · R(r ) = Im n− 1 2 n = 1, 2, · · · ∞ ∞ u (r, θ, z ) = m=1 n=1 cmn Im n− 1 2 1 π r sin m θ sin n − H 2 π z H at r = a ∞ ∞ β (θ, z ) = m=1 n=1 n− cmn Im 1 2 π 1 a sin m θ sin n − H 2 coefficient of expansion cmn = π 0 Im n− H 0 β (θ, z ) sin m θ sin n − 1 2 π H a π 0 H 0 1 2 π H z dz dθ sin2 m θ sin2 n − 120 1 2 π H z dz dθ π z H π r H R ...
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