Let consider the following equation on $[0,T]\times \mathbb{R}$: $$ \partial_t u + \beta \partial_x u = \alpha \partial_{xx} u. $$
Question: Is it possible to construct a fundamental solution for this equation ?
I was thinking in reformulate it under the form: $$ \partial_t u + \partial_x ( \beta u - \alpha \partial_{x} u ) = 0. $$
What strategy do you think is the appropriate to obtain a Green-like function for this equation ?