What really are perturbation expansions? I'm unsure if this question belongs here or at Math.SE, but since I've got to it by reading some articles about Physics I'm going to post it here anyway.
In this particular article (Theoretical models in low-Reynolds-number locomotion) about fluid mechanics I've found the following situation: one gets to solve Stokes' equations. The equations themselves are linear, but there are still a problem with the boundary conditions which may be evaluated on some weird surface.
In that particular article the author solved the problem with a perturbation expansion. Basically he chose a parameter $\epsilon$ and wrote the solution $\psi$ as
$$\psi = \epsilon \psi_1 + \epsilon^2 \psi_2 + \cdots$$
and as I understood, $\psi_n$ is the solution to the problem with $O(\epsilon^n)$ boundary conditions.
This seems to be something that is quite frequently done. One expands some function in a perturbation series like that using a parameter. The problem is that I can't get what this really is.
This seems quite different than expanding the general solution of a differential equation in a certain basis of functions. There's also this parameter $\epsilon$ that I can't get his role on all of this.
So what really is this perturbation expansion? From a rigorous point of view what is that series? And why is it useful anyway?
 A: Think of this not as an extremely rigorous way of solving the differential equation, but rather as using your intuition to guess a solution.  Often when you are given a differential equation, the solution is not at all obvious, and perhaps the equation isn't even solvable analytically.  Instead of giving up, though, sometimes you can identify a parameter (the $\epsilon$ in your above expansion) such that for $\epsilon=0$, the equation is easy to solve.  You can then guess that as long as $\epsilon$ is "small," you can Taylor expand about the $\epsilon=0$ solution to get a perturbative series solution to the true problem.  You then hope that this series will converge for the actual value of $\epsilon$, and this will give you your actual solution.
Physically, this sort of thing happens often when you have a system that is "close" to some special system that is easy to solve.  Maybe you have some sort of oscillator with energy $E$ that is "pretty close" to being a simple harmonic oscillator (which is very well understood), but whose potential differs from a true harmonic potential by some factor $\epsilon V$ where $V$ is of the same order of magnitude as $E$ and $\epsilon$ is small (so $\epsilon V \ll E$).  Then it is reasonable to expect that the behavior of this system should be "pretty close" to the behavior of the simple harmonic oscillator, and that as you vary $\epsilon$ in some neighborhood of $0$, the system's behavior should change smoothly.  But this means that you should be able to expand the general system's solution as a Taylor series in $\epsilon$ with the zeroth-order term simply being the solution for a simple harmonic oscillator and higher-order terms giving corrections proportional to powers of $\epsilon$.
