ps8sol - ACM 95/100c Problem Set 8 Solutions Lei Zhang May...

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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 Poissons equation inside a circle of radius a : 2 u = f. with boundary conditions u ( a, ) = h 1 ( ) for 0 < < u r ( a, ) = h 2 ( ) for- < < Represent the solution u ( r, ) in terms of the Greens function which you can assume to be known but indicate what equation is satisfied by the Greens function and the boundary conditions for the Greens function as well. Solution 1 the boundary condition for u is u ( a, ) = h 1 ( ) for < < (1) and r u ( a, ) = h 2 ( ) for- < < (2) we have Greens 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 Greens function as 2 G = ( x- x ) (4) with boundary condition G ( a, ) = 0 for < < (5) and r G ( a, ) = 0 for- < < (6) therefore, the solution can be represented as u ( x ) = Z Z A Gfdx + a Z h 1 ( ) r G ( a, ) d- a Z- G ( a, ) h 2 ( ) d (7) 1 Problem 2 (No collaboration) Determine the Greens function G ( x , x ) inside a sphere of radius a for Poissons equation with Dirichlet boundary conditions given on the surface of the sphere....
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ps8sol - ACM 95/100c Problem Set 8 Solutions Lei Zhang May...

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