Derive the corresponding odes

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Reference no: EM131004005

Consider Laplace equation,

(∂2u / ∂x2) + (∂2u / ∂y2) = 0,

over the annulus in R2, {(r, θ) | 1 < r < 2, -Π ≤ θ < Π} in the standard polar coordinates.

a. Prove that in polar coordinates, Laplace equation is

(∂2u / ∂r2) + (∂u / r ∂r) + (∂2u / r2∂θ2) = 0,

b. Let u(r, θ) = R(r) - Θ(θ) and derive the corresponding ODEs for R(r) and Θ(θ).

c. Assume the boundary conditions on u,

u(r, -Π)= u(r, Π), and (∂u / ∂θ) (r, -Π) = (∂u / ∂θ) (r, Π)

Find the corresponding conditions on Θ(θ). Then solve for Θ and R, and write down the general solution of u(r, θ) in an infinite series.

d. Further assume boundary conditions u(1, θ) = 0 and u(2, θ) = cos θ. Find the solution u(r, θ).

Reference no: EM131004005

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