Differential Equations Lecture Work Solutions 108

Differential Equations Lecture Work Solutions 108 - 1 . u t...

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1. u t = k 2 u 0 r a u ( a, θ, t )=0 0 θ 2 π u ( r, θ, 0) = α ( r, θ ) t> 0 ˙ TR Θ= kT h Θ 1 r ( rR 0 ) 0 + 1 r 2 R Θ 0 i ˙ T kT = 1 Rr ( rR 0 ) 0 + 1 r 2 Θ 0 Θ = λ ˙ T + kλT =0 r R ( rR 0 ) 0 + Θ 0 Θ = λr 2 r R ( rR 0 ) 0 + λr 2 = Θ 0 Θ = µ Θ 0 + µ Θ=0 r ( rR 0 ) 0 +( λr 2 µ ) R =0 Θ(0) = Θ(2 π ) | R (0) | < Θ 0 (0) = Θ 0 (2 π ) R ( a )=0 ⇓⇓ µ m = m 2 R m = C 1 m J m ( λr )tosat is fy | R (0) | < Θ m = sin m
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