Reference no: EM132400078

Question 1. Given the following Green's function problem on a unit disc

∇²g = -δ(r - r'), 0 < r, r' < 1, 0 < θ, θ' < 2Π, g(1, θ; r', θ') = 0,

the method of images can be used to calculate the following Green's function

g(r; r') = -1/2Π.ln(|r - r'|/|r'||r - r*|), where r* = r'/|r'|²

Here, r* is the image of r' outside of the disc with the reciprocal distance from the origin. Show that this Green's function satisfies the boundary condition for any point P on the boundary.

Question 2. Consider the following boundary value problem in the quarter plane (x, y > 0)

∇²u = - f(x, y), u(0, y) = b(y), ∂u/∂y (x, 0) = 0

(a) Determine the corresponding Green's function problem.
(b) Express the solution u(x, y) in terms of a Green's function.
(c) Use the method of images to determine the Green's function

Question 3. Given a Dirichlet problem for the Poisson equation in an annulus, where in polar coordinates, the domain is defined as θ ∈ [0, 2Π] with the radius having the bounds r ∈ [a, b], the correspomding Green's function problem is given by

∇²g = 1/r. ∂/∂r(r∂g/dr) + 1/r² ∂²g/∂θ² = -1/r δ(r -r')δ(θ - θ'), a < r, r' < b, 0 < θ, θ' < 2Π

g(a,θ;r',θ') = g(b,θ;r',θ') = 0, g(r,0;r',θ')= g(r,2Π;r',θ'), ∂g/∂θ(r,0;r',θ') = ∂g/∂θ (r, 2Π; r', θ')

(a) Determine the θ-dependent eigenfunctions φn(θ) for a partial eigenfunction expansion.
(b) What is the resulting Green's function problem for the expansion coefficients g_n(r; r', θ')?
(c) Solve for the resulting coefficients and express the Green's function as a sum.