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Differential Equations Lecture Work Solutions 114

Differential Equations Lecture Work Solutions 114 - subject...

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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, 0 < z < H
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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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