Singular Perturbation

In mathematics, more precisely in perturbation theory, a singular perturbation problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to zero. This is in contrast to regular perturbation problems, for which an approximation can be obtained by simply setting the small parameter to zero.

More precisely, the solution cannot be uniformly approximated by an asymptotic expansion

as . Here is the small parameter of the problem and are a sequence of functions of of increasing order, such as . 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.