Differential Equations Lecture Work Solutions 129

Differential Equations Lecture Work Solutions 129 - 2. + =...

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2. ϕ 0 + µϕ =0 ϕ (0) = ϕ ( π )=0 µ m = m 2 ϕ m =s in m =1 , 2 , ··· u ( r, θ, ϕ )= X n =0 X m =1 r n P m n (cos θ ) | {z } Θ equation A nm sin Requat ion u ( a, θ, ϕ )= f ( θ, ϕ )= X n =0 X m =1 a n A nm P m n (cos θ )s in a n A nm = R π 0 R π 0 f ( θ, ϕ ) P m n (cos θ )s in sin θdθdϕ | {z } area elem. R π 0 R π 0 ( P m n (cos θ )) 2 sin 2 sin θdθdϕ A nm = R π 0 R π 0 f ( θ, ϕ ) P m
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