solh6 - Solution to homework Set 6: Math 716 1. Determine...

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Unformatted text preview: Solution to homework Set 6: Math 716 1. Determine the Green’s function G ( x , x ) for Dirichlet condition for Laplace’s equation in 3-D dimensions in the hemispherical domain: Ω = { x : | x | < 1 , x 3 > } Use this to determine an integral expression for u ( r, θ, φ ) in spherical polar-coordinates satisfying Δ u = 0 inΩ , with u = 0 for θ = 0 , and u (1 , θ, φ ) = ψ ( θ, φ ) for θ ∈ [0 , π 2 ] , φ ∈ [0 , 2 π ] Solution: Note that if x = ( x , 1 , x , 2 , x , 3 ) be the location of a unit positive charge inside the hemisphere, to satisfy the boundary condition at x 3 = 0, we need a negative unit charge at ¯ x = ( x , 1 , x , 2 ,- x , 3 ). The image of positive charge x outside the unit sphere is a negative unit charge at x | x | 2 , while the image of negative unit charge at ¯ x outside the unit sphere is a positive unit charge at ¯ x | ¯ x | 2 . Combining, we have G ( x ; x ) =- 1 4 π ( 1 | x- x |- 1 | x- ¯ x |- 1 | x- x | x | 2 | + 1 | x- ¯ x | ¯ x | 2 | ) We note that on | x | = 1, ∂ ∂n- 1 | x- x | = x · ∇ 1 | x- x | = x · ( x- x ) | x- x | 3 If x = ( r , θ , φ ), and on the spherical surface x = (1 , θ, φ ), then x · ( x- x ) = 1- r cos Ψ, where Ψ is the angle between x and x and is given by (using dot products): cos Ψ = sin θ sin θ cos φ cos φ + sin θ sin θ sin φ sin φ + cos θ cos θ = sin θ sin θ cos( φ- φ ) + cos θ cos θ Therefore, ∂ ∂n- 1 | x- x | = 1- r cos Ψ (1 + r 2- 2 r cos Ψ) 3 / 2 We note that the image point ¯ x has spherical coordinates (...
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This note was uploaded on 07/26/2011 for the course MATH 716 taught by Professor Tanveer,s during the Spring '08 term at Ohio State.

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solh6 - Solution to homework Set 6: Math 716 1. Determine...

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