Differential Equations Lecture Work Solutions 116

Differential Equations Lecture Work Solutions 116 - 1 1...

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1. 1 r ( ru r ) r + 1 r 2 u θθ + u zz =0 (a) Θ 0 + µ Θ=0 Z 0 λZ =0 r ( rR 0 ) 0 +( λr 2 µ ) R =0 Θ(0) = Θ(2 π ) Z ( H )=0 | R (0) | < Θ 0 (0) = Θ 0 (2 π ) R ( a )=0 ⇓⇓ µ 0 =0 R nm = J m ( λ nm r ) Θ 0 = 1 satisfes boundedness µ m = m 2 Θ m = sin cos J m ( λ nm a )=0 yields eigenvalues m =1 , 2 , ··· n =1 , 2 , ··· λ> 0! Z nm = sinh ± λ nm ( z H ) vanishes at z = H u ( r, θ, z )= X m =0 X n =1 ( a nm cos + b nm sin )s inh ± λ nm ( z H ) J m ( ± λ nm r ) This is zero For m =0 α ( r, θ )= X m =0 X n =1 ( a nm cos + b nm sin )s inh ± λ nm ( H ) | {z } this is a constant J m ( ± λ nm r ) a nm = R a 0 R 2
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