3. Green's functions (Haberman § 9.3, see problems 9.3.9 and 9.3.11)

Consider

d'u

dr²+u = f(x)

subject to

subject to

u(0) = 0, u(x/2) = 0.

(14)

The goal in (a) is to find an integral representation for the unknown u(r) of the

form

u(x) = ™² G(E,x)ƒ (E)d£

(15)

where G(r, ) is the Green's function. Note that (15) only holds for homogeneous

boundary conditions (e.g. (14)).

(a) Solve for G(§, z) directly from

JG (§, x)

მ2

(13)

+G(§, x) = 8(§ - x)

(16)

G(0,r)=0

G(T/2, x) = 0.

(17)

You will need to determine and apply the matching conditions at = r as

discussed in lecture to find G(z, E) (see also Haberman page 388).