homework6 - n = 3 then there exists a constant C such that...

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Homework Set 6: Math 716, Due Friday, March 6th 1. Determine the Green’s function G ( x , x 0 ) for Dirichlet condition for Laplace’s equation in 3-D dimensions in the hemispherical domain: Ω = { x : | x | < 1 , x 3 > 0 } 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 ] [0 , 2 π ] Hint: Better not to use spherical polar coordinates until after you get an integral repre- sentation. 2. If S ( x ,t ) is the source solution to the heat equation given by S ( x ,t ) = p 1 4 πκt P n/ 2 exp b - | x | 2 4 κt B , then show that R ( x ,t ; x 0 ,t 0 ) = S ( x - x 0 ,t - t 0 ) for t > t 0 and R = 0 for t < t 0 x R n satisFes R t - κ Δ R = δ ( x - x 0 ) δ ( t - t 0 ) 3. Consider the wave equation u tt - Δ u = 0 for x R n , t > 0 , with u ( x , 0) = φ ( x ) ,u t ( x , 0) = ψ ( x ) , with compactly supported φ and ψ . a. Show that for
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Unformatted text preview: n = 3, then there exists a constant C such that | u ( x ,t ) | < C 1 + t for x ∈ R 3 , t > b. Show that for n = 3, at a Fxed point x outside the support of φ and ψ , there exists an earliest impact time t i > 0 that depends on x so that u ( x ,t ) = 0 , for 0 ≤ t < t i , and that there also exists a Fnal impact time t f > t i so that for t > t f , u ( x ,t ) = 0. c. Show that for n = 2, for x outside the support of ψ and φ , there exists an initial impact time t i > 0 depending on x but no Fnal Fnal impact time t f . Hint: Examine carefully the consequences of Poisson and Kircho± formula. 1...
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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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