Applying Fourier transform + Green's function on diffusion equation question

Question

I stumbled upon this question which says;

The following partial differential equation is called the diffusion equation:

The function ϕ(x,t) represents (for example) the number density of a gas species as a function of position x and time t. The “source function” f(x,t) describes the rate at which atoms of the gas are emitted at each point of space and time. The “diffusion constant” D describes how quickly the atoms diffuse through space.

a) The Green’s function is a solution to the partial differential equation

Derive the relationship between ϕ(x,t) and G(x − x',t − t').

(b) Let us Fourier transform G in the space coordinate (but not in time): . (3)

Derive the ordinary differential equation (ODE) in t satisfied by G(k, t − t').

I have no clue how should I start with, I really need help and guidance. Thanks...

Answer 1

$\phi(x,t) = G(x-x', t-t')$ only if $f(x,t) = \delta(x-x')\delta(t-t')$.

But you are indeed in the right neighborhood. Hint:

$$f(x,t) = \int\int f(x',t') \delta(x- x')\delta(t-t')dx'dt'.$$