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MATHS 361: Partial Differential Equation

Tutorial 11: Green’s function

1. Suppose I have the equation:

uxx + ux = ex

with the boundary conditions ux(0) = 0 and u(1) = 0.

How would I solve this using Green’s functions?

2. Suppose I have the equation:

uxx + u2 = sin(x)

with the boundary conditions ux(0) = 0 and u(1) = 0.

How would I solve this using Green’s functions?

3. Suppose I have the equation:

uxx + u = x

with the boundary conditions ux(0) = 4 and u(π) = − 1.

How would I solve this using Green’s functions?

4. Suppose I have boundary conditions ux(0) = 0 and u(1) = 0.

Can you construct an Linear operator L[u] such that these boundary conditions are forbidden.

5. I am in domain depicted above:  my function is in 2D, and I seek solutions to the equation ∆u = δ(x − 1,y − 0.2) within a wedge shaped domain (the angle of the wedge is 60 degrees, a.k.a.  π/3 radians). For this question, please give your answers as diagrams, rather than explicit calculations.

(a) Assuming I have reflecting boundary conditions on both my edges, is it possible to solve via

reflections? If it is possible, where would I place my δ spikes on the plane, and what sign would they be? If it is not possible, what goes wrong?

(b) Assuming I have absorbing boundary conditions on both my edges is it possible to solve via

reflections? If it is possible, where would I place my δ spikes on the plane, and what sign would they be? If it is not possible, what goes wrong?

(c) Assuming I have one absorbing and one reflecting boundary condition is it possible to solve via reflections? If it is possible, where would I place my δ spikes on the plane, and what sign would they be? If it is not possible, what goes wrong?