Reference no: EM132400081

Question 1. Consider the following Poisson equation on the upper half of a unit disc:

∇²u(x, y) = -f(x, y), Ω = {x, y | x² + y² ≤ 1, 0 ≤ y}, u(x, y)|_∂Ω = b(x, y)

(a) Use Green's second identity to derive a corresponding Green's function problem (in either Cartesian or polar coordinates), and express the solution u in terms of the Green's function.

(b) Determine the Green's function using a partial eigenfunction expansion. Note that the boundary conditions in θ are no longer periodic, so the eigenfunctions and othogonality conditions will change.

(c) Now, determine the Green's function using the method of images.

(d) Let f (x, y) = 0, b(x, y) = 0 along the top of the half-disc (i.e., when r = 1), and b(x , y) = 1 along the bottom of the half-disc (i.e., when y = 0). Use either Green's function to express the solution u(x, y) in either coordinate system. Evaluate any derivatives, but you do NOT need to evaluate integrals.

Question 2. Consider the following Green's function on a horizontal infinite strip of width Π:

∇²g(x,y;x',y') = -δ(x - x')δ(y - y'), Ω = {x,y|x ∈ R, 0 ≤ y ≤ Π},
g(x, 0; x', y') = 0, g(x, Π; x', y') = 0, lim_|x|→∞ g(x, y; x', y') = 0

(a) Use a partial eigenfunction expansion by expanding the y-component of the differential operator.

(b) Determine the resulting Green's function problem for the expansion coefficients gm(x; x', y'), and use a Fourier transform in x to calculate g^{^}_n(k; x', y') in Fourier space.

(c) Invert on g^{^}_n(k; x', y') back to real space. The integral ∫_Rcos(ax)/(x²+b²).dx = Π/|b|e^-|a||b| might help.

(d) Now, determine the Green's function using conformal mapping. Hint: the function f(z) = e^z maps Ω to the upper-half plane.