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

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.

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