Consider the following advection problem with weak diffusion: $$ \varepsilon\partial_{x}^2 u=\partial_{t}u+\partial_{x}u, $$ for $−\infty < x < \infty$, and $t > 0$ where $u(x, 0) = f(x)$.
- Using multiple scales, find a first-term approximation of the solution that is valid for large t.
- Try to solve the original problem directly(Hint: The Fourier transform is a good choice).
I have got the second question's answer, but I can not find a kind of appropriate scales for the first question.