I was reading an article about pertubation in advection-transport equations, nad so they have defined the following equation with the perturbation ($\epsilon). $
$$y_t+-\epsilon .y_{xx}+ M.y_x=0\, ;(x,t) \in (0,1)\times(0,T)$$ $$y(0,t)=v(t), \text{if} \, M>0 \, ;t\in (0,T)$$
$$y(1,t)=0 \,\text{if}\,M<0 \, ;t\in (0,T)$$ $$y(x,0)=y_0(x) \, ; x\in (0,1)$$ And they took the case of$\epsilon=0 $, we will get the transport equation. $$y_t+M.y_x=0\, ;(x,t) \in (0,1)\times(0,T)$$ $$y(0,t)=v(t), \text{if} \, M>0 \, ;t\in (0,T)$$
$$y(1,t)=0 \,\text{if}\,M<0 \, ;t\in (0,T)$$ $$y(x,0)=y_0(x) \, ; x\in (0,1)$$
and said that we have two boundary layers :
1/ In $x=1$ of size $O(\epsilon)$
2/ In the characterestic $\{(x,t)\in(0,1)\times (0,T): x-M.t=0\}$ of size $O(\sqrt{\epsilon})$
And I did not get how they concluded these results.
This is a partial answer, the details are in the linked paper, and references therein, I'm not going to attempt to rewrite it.
With $\epsilon=0$ the equation is order 1 in space, so can't satisfy 2 general boundary conditions. To find the location/size of the boundary layer, you can rescale $\eta=(x-x_0)/\epsilon^\alpha$ to give $$ y_t-\epsilon^{1-2\alpha}y_{\eta\eta}+\epsilon^{-\alpha}My_\eta=0 $$ and so if $\alpha=1$ you have a dominant balance, so there is a boundary layer width $O(\epsilon)$.
If $M>1$ the boundary layer must be at $x_0=1$, and if $M<0$ it will be at $x_0=0$. The paper you linked to does go through these calculations in limited detail for the $M>0$ case.
Section 2 of the paper gives some more details about the different substitutions.
Also see this answer for general information about finding the location/width of boundary layers.