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Differential Equations Lecture Work Solutions 109

Differential Equations Lecture Work Solutions 109 - u(r t =...

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u ( r, θ, t ) = m =1 n =1 ( a nm cos m θ + b nm sin m θ ) e k λ nm t J m ( λ nm r ) + n =1 a n 0 · 1 0 e k λ n 0 t J 0 ( λ n 0 r ) m =0 α ( r, θ ) = n =1 a n 0 J 0 ( λ n 0 r ) + m =1 n =1 ( a nm cos m θ + b nm sin m θ ) J m ( λ nm r ) a n 0 = 2 π 0 a 0 α ( r, θ ) J 0 ( λ n 0 r ) rd rdθ 2 π 0 a 0 J 2 0 ( λ n 0 r ) rd rdθ a nm = 2 π 0 a 0 α ( r, θ ) cos m θ J m ( λ nm r ) rd rdθ 2 π 0 a 0 cos 2 m θ J 2 m ( λ nm r ) rd rdθ b nm = 2 π 0 a
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