Suppose I have a PDE:
$$\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\left(k(x)\frac{\partial T}{\partial x}\right)+Q(t)$$
with the following initial/boundary conditions:
$T(0,x)=0$ and $T(t,0)=T(t,L)=0$.
Suppose that $k(x)$ is a smooth function for the time being. I'm looking for a Green's function for my equation.
Ideally I want to use this forward problem to formulate an inverse problem to find either $k(x)$ or $Q(t)$ from data in the form on: $$T_{av}(t)=\int_{0}^{L}T(t,x)dx $$