A very brief introduction to perturbation methods

Hi everybody


Today I'm going to talk about a topic which in my opinion it's fascinating and very imporant to be known for all those who want to practice cutting edge science. I'm refering to perturbation methods.


These mathematical techniques were developed approximately one century ago by scientist focussed on dynamic systems mathematics or working on engineering and physics where several non analytical solvable systems of equations are the base of development to their respective areas (for example, fluid mechanics).


Perturbation methods, also called asymptotic, deal with different ways to get very valuable information of a system without solving their equations. These techniques introduce which is called "perturbed solutions": a solution is assumed depending of a very small variable (usually is called $\epsilon$) and other desired parameters. The structure of these solutions depends of which type of expansion we are planning to introduce. Typical expansions are for example power series. If U is a solution of the equation, then we could assume that:


$$U=U_0+\epsilon^m U_1 + \epsilon^n U_2 + ...$$

where $U_1$,$U_2$, etc. are solutions that appear when we assume order $\epsilon^m$, $\epsilon^n$, etc.

As this solution satisfy the equation, we can just substitute on it and we can obtain a system of equations at different $|epsilon$ orders that we can solve them to get the asymptotic approximation. We can also perform a nonlinear dynamic study to get stability values, nodes, etc.

As it can be seen, perturbative methods are extremely useful, but its application is not so immediate. Use this method with differential equations is a very hard task, even impossible for those not accostumed to work with this type of methods. That's the reason for which only few people in the world are really experts in this field.

In the following posts I will explain a little bit more these kind of methods, and some examples of application.