Differential Equations Lecture Work Solutions 114

Differential - subject to the boundary conditions u r θ 0 = 0 u z r θ H = 0 u r z = u r π z = 0 u a θ z = β θ z 3 Find the solution to the

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4.6 Laplace’s Equation in a Circular Cylinder Problems 1. Solve Laplace’s equation 1 r ( ru r ) r + 1 r 2 u θθ + u zz =0 , 0 r<a , 0 <θ< 2 π, 0 <z<H subject to each of the boundary conditions a. u ( r, θ, 0) = α ( r, θ ) u ( r, θ, H )= u ( a, θ, z )=0 b. u ( r, θ, 0) = u ( r, θ, H )=0 u r ( a, θ, z )= γ ( θ, z ) c. u z ( r, θ, 0) = α ( r, θ ) u ( r, θ, H )= u ( a, θ, z )=0 d. u ( r, θ, 0) = u z ( r, θ, H )=0 u r ( a, θ, z )= γ ( z ) 2. Solve Laplace’s equation 1 r ( ru r ) r + 1 r 2 u θθ + u zz =0 , 0 r<a , 0 <θ<π,
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Unformatted text preview: subject to the boundary conditions u ( r, θ, 0) = 0 , u z ( r, θ, H ) = 0 , u ( r, , z ) = u ( r, π, z ) = 0 , u ( a, θ, z ) = β ( θ, z ) . 3. Find the solution to the following steady state heat conduction problem in a box ∇ 2 u = 0 , ≤ x < L, < y < L, < z < W, subject to the boundary conditions 114...
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This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.

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