What is the green function for:
$$\frac{\partial u }{\partial t}-\frac{\partial u }{\partial x}-\delta \left ( x \right )u=f(x,t)$$ Initial condition is zero and boundary condition is zero at infinity. The domain is from 0 to +$\infty$. I got Laplace transform first to get: $$su-\frac{\partial u }{\partial x}-\delta \left ( x \right )u=\delta \left ( x-\zeta \right )e^{-s\tau }$$