I am wondering if Green function exists for heat conduction and convection in a slab. The problem can be expressed as, $$ \frac{\partial T}{\partial t} = k\frac{\partial ^2T}{\partial x^2}-v\frac{\partial T}{\partial x} $$
Boundary condition: $$ \frac{\partial T}{\partial x}=g(t), x = 0, t>0 $$ $$ \frac{\partial T}{\partial x}=0, x = L, t>0 $$
If no Green function exists for such a problem, how about the semi-infinite media by changing the right boundary at $L$ to $\infty$. I know the Green functions for heat conduction problem with arbitrary boundary condition are well documented, but the convection problem is not. I found some literature that gave the Green function in frequency domain considering the heat convection, but I didn't find a closed form in temporal domain. I appreciate for any comments on this problem. Thanks in advance.