I'm trying to solve $\theta_t - D\theta_{xx} = f(x,t)$ on the half-line $0 < x < \infty$ for $0< t < \infty$ given boundary and initial conditions $\theta(0,t) = h(t)$, $\theta(x,0) = \Theta(x)$. I've been told to consider $V(x,t) = \theta(x,t) - h(t)$ and use the method of images.
I've got that $V_t - DV_{xx} = f(x,t) - h'(t)$, $V(0,t) = 0$, $V(x,0) = \theta(x) - h(0)$.
What should I do now? My only idea is to find the Green's function satisfying
$\frac{\partial G}{\partial t} - D \frac{\partial^2 G}{\partial x^2} = \delta(t - \tau)\delta(x - \zeta)$
but I don't know how to do this.