Differential Equations Lecture Work Solutions 127

Differential Equations Lecture Work Solutions 127 - 2 u...

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4.7 Laplace’s equation in a sphere Problems 1. Solve Laplace’s equation on the sphere u rr + 2 r u r + 1 r 2 u θθ + cot θ r 2 u θ + 1 r 2 sin 2 θ u ϕϕ = 0 , 0 r < a, 0 < θ < π, 0 < ϕ < 2 π, subject to the boundary condition u r ( a, θ, ϕ ) = f ( θ ) . 2. Solve Laplace’s equation on the half sphere u rr + 2 r u r + 1 r 2 u
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Unformatted text preview: 2 u θθ + cot θ r 2 u θ + 1 r 2 sin 2 θ u ϕϕ = 0 , ≤ r < a, < θ < π, < ϕ < π, subject to the boundary conditions u ( a, θ, ϕ ) = f ( θ, ϕ ) , u ( r, θ, 0) = u ( r, θ, π ) = 0 . 127...
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