When available, explicit solutions are fundamental to the understanding of differential equations, but for many problems of interest, such closed-form exact solutions can not be found. Even when they do exist, often the understanding and interpretation of explicit solutions requires computational and/or further approximation. The study of asymptotic and perturbation methods offers a systematic analysis of solutions to differential equations when tractable solutions are unavailable. The goal of perturbation methods is to replace an exact problem that does not have a tractable solution, by a series of approximate problems that do have (simpler) explicit solutions. We will cover a range of topics including asymptotic expansions for algebraic and differential equations and multiple scales for initial and boundary value problems. The methods all rely on there being a parameter in the problem that is relatively small. Such a situation is relatively common in applications and this is one of the reasons that perturbation methods are a cornerstone of applied mathematics.
Introduction to Perturbation Methods by M.H. Holmes