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s4_95c

# s4_95c - u r θ ϕ 0 = f r θ ϕ ∂u ∂t r θ ϕ 0 = 0...

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ACM 95/100c April 23, 2004 Problem Set IV 0.- (2 pts) Write down your grading-section number. 1.- (30 pts) Find the solution of Laplace’s equation 2 u = u xx + u yy + u zz = 0 in the sphere ρ = x 2 + y 2 + z 2 1 subject to each one of the following boundary conditions (a) (10 pts) u = x 3 for ρ = 1. (b) (10 pts) u = x 2 + y 2 for ρ = 1. (c) (10 pts) u = e z for ρ = 1. 2.- (5 pts) Let u be a function defined in the sphere S a = { ρ a } . Denoting by n the outward unit normal to S a , show that on S a we have ∂u ∂n = n · ∇ u = ∂u ∂ρ . 3.- (20 pts) Solve the following heat conduction problem in the sphere: ∂u ∂t - ∇ 2 u = 0 for ρ < 1 , t > 0 , ∂u ∂ρ = 0 for ρ = 1 , t > 0 , u ( r, θ, ϕ, 0) = f ( r, θ, ϕ ) . 4.- (20 pts) Solve the following vibration problem in the sphere: 2 u ∂t 2 - ∇ 2 u = 0 for ρ < 1 , t > 0 , u = 0 for ρ = 1 , t >

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Unformatted text preview: , u ( r, θ, ϕ, 0) = f ( r, θ, ϕ ) ∂u ∂t ( r, θ, ϕ, 0) = 0 . 5.- (25 pts) (a) (15 pts) Solve ∇ 2 u = 0 for ρ < 1 , t > , ∂u ∂ρ = g ( θ, φ ) for ρ = 1 where the function g is assumed to satisfy Z 2 π Z π g ( θ, ϕ ) sin( θ ) dθdϕ = 0 . (1) Show that this problem admits a unique solution up to an additive constant, that can be determined if, for example, we assume the temperature scale is such that u = 0 at ρ = 0 for t = 0. (b) (10 pts) What happens if the integral in (1) is not zero? Compare with problem 1.: no special conditions on the boundary values were needed there. Explain! Due April 30, 3:00pm....
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