Singular perturbation

In mathematics, a singular perturbation problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to zero. More precisely, the solution cannot be uniformly approximated by an asymptotic expansion

${\displaystyle \varphi (x)\approx \sum _{n=0}^{N}\delta _{n}(\varepsilon )\psi _{n}(x)\,}$

as ${\displaystyle \varepsilon \to 0}$. Here ${\displaystyle \varepsilon }$ is the small parameter of the problem and ${\displaystyle \delta _{n}(\varepsilon )}$ are a sequence of functions of ${\displaystyle \varepsilon }$ of increasing order, such as ${\displaystyle \delta _{n}(\varepsilon )=\varepsilon ^{n}}$. This is in contrast to regular perturbation problems, for which a uniform approximation of this form can be obtained. Singularly perturbed problems are generally characterized by dynamics operating on multiple scales. Several classes of singular perturbations are outlined below.

The term "singular perturbation" was coined in the 1940s by Kurt Otto Friedrichs and Wolfgang R. Wasow.^[1]