Singular perturbation

Remarks

Example: $P_\varepsilon$ is $\varepsilon y'' + y' = 0$ on $D = [0, 1]$ with $y(0) = 0$ and $y(1) = 1$. The solution is $y_\varepsilon(x) = (1 - e^{-x/\varepsilon}) / (1 - e^{-1/\varepsilon})$. The reduced problem $y' = 0$, $y(0) = 0$, $y(1) = 1$ has no solution. Indeed $y_\varepsilon(x) \to 1$ for each $x \in (0, 1]$ while $y_\varepsilon(0) = 0$, so the convergence is not uniform.

Non-example: $y' = -y + \varepsilon$ on $[0, 1]$ with $y(0) = 1$. The solution $y_\varepsilon(x) = \varepsilon + (1 - \varepsilon) e^{-x}$ converges uniformly to $e^{-x}$, the solution of the reduced problem, so the perturbation is regular.

A small parameter multiplying the highest derivative usually makes the problem singular. Singular problems are treated by the method of matched asymptotic expansions and the method of multiple scales.