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Unformatted text preview: ACM 95/100c Problem Set 8 Solutions Lei Zhang May 23, 2006 Each Problem is worth 10 points. Each part of a multi-part problem is weighted equally. Problem 1 (No collaboration) Consider the solution of Poisson’s equation inside a circle of radius a : ∇ 2 u = f. with boundary conditions u ( a,θ ) = h 1 ( θ ) for 0 < θ < π ∂u ∂r ( a,θ ) = h 2 ( θ ) for- π < θ < Represent the solution u ( r,θ ) in terms of the Green’s function which you can assume to be known but indicate what equation is satisfied by the Green’s function and the boundary conditions for the Green’s function as well. Solution 1 the boundary condition for u is u ( a,θ ) = h 1 ( θ ) for < θ < π (1) and ∂ ∂r u ( a,θ ) = h 2 ( θ ) for- π < θ < (2) we have Green’s formula for u and G , Z Z ( u ∇ 2 G- G ∇ 2 u ) dA = I ( u ∇ G- G ∇ u ) · n ds (3) therefore, we can define the Green’s function as ∇ 2 G = δ ( x- x ) (4) with boundary condition G ( a,θ ) = 0 for < θ < π (5) and ∂ ∂r G ( a,θ ) = 0 for- π < θ < (6) therefore, the solution can be represented as u ( x ) = Z Z A Gfdx + a Z π h 1 ( θ ) ∂ ∂r G ( a,θ ) dθ- a Z- π G ( a,θ ) h 2 ( θ ) dθ (7) 1 Problem 2 (No collaboration) Determine the Green’s function G ( x , x ) inside a sphere of radius a for Poisson’s equation with Dirichlet boundary conditions given on the surface of the sphere....
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- Spring '09
- Boundary value problem, 2 g, Boundary conditions, Dirichlet boundary condition, 0 g, boundary condition