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
φ ( x ) ≈ ∑ n = 0 N δ n ( ε ) ψ n ( x ) {\displaystyle \varphi (x)\approx \sum _{n=0}^{N}\delta _{n}(\varepsilon )\psi _{n}(x)\,}
as ε → 0 {\displaystyle \varepsilon \to 0} . Here ε {\displaystyle \varepsilon } is the small parameter of the problem and δ n ( ε ) {\displaystyle \delta _{n}(\varepsilon )} are a sequence of functions of ε {\displaystyle \varepsilon } of increasing order, such as δ n ( ε ) = ε n {\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.
Methods of analysis A perturbed problem whose solution can be approximated on the whole problem domain, whether space or time, by a single asymptotic expansion has a regular perturbation. Most often in applications, an acceptable approximation to a regularly perturbed problem is found by simply replacing the small parameter ε {\displaystyle \varepsilon } by zero everywhere in the problem statement. This corresponds to taking only the first term of the expansion, yielding an approximation that converges, perhaps slowly, to the true solution as ε {\displaystyle \varepsilon } decreases. The solution to a singularly perturbed problem cannot be approximated in this way: As seen in the examples below, a singular perturbation generally occurs when a problem's small parameter multiplies its highest operator. Thus naively taking the parameter to be zero changes the very nature of the problem. In the case of differential equations, boundary conditions cannot be satisfied; in algebraic equations, the possible number of solutions is decreased. Singular perturbation theory is a rich and ongoing area of exploration for mathematicians, physicists, and other researchers. The methods used to tackle problems in this field are many. The more basic of these include the method of matched asymptotic expansions and WKB approximation for spatial problems, and in time, the Poincaré–Lindstedt method, the method of multiple scales and periodic averaging. The numerical methods for solving singular perturbation problems are also very popular. For books on singular perturbation in ODE and PDE's, see for example Holmes, Introduction to Perturbation Methods, Hinch, Perturbation methods or Bender and Orszag, Advanced Mathematical Methods for Scientists and Engineers.
Examples of singular perturbative problems Each of the examples described below shows how a naive perturbation analysis, which assumes that the problem is regular instead of singular, will fail. Some show how the problem may be solved by more sophisticated singular methods.
Vanishing coefficients in ordinary differential equations Differential equations that contain a small parameter that premultiplies the highest order term typically exhibit boundary layers, so that the solution evolves in two different scales. For example, consider the boundary value problem
ε u ′ ′ ( x ) + u ′ ( x ) = − e − x , 0 < x < 1 u ( 0 ) = 0 , u ( 1 ) = 1. {\displaystyle {\begin{matrix}\varepsilon u^{\prime \prime }(x)+u^{\prime }(x)=-e^{-x},\ \ 0<x<1\\u(0)=0,\ \ u(1)=1.\end{matrix}}}
Its solution when ε = 0.1 {\displaystyle \varepsilon =0.1} is the solid curve shown below. Note that the solution changes rapidly near the origin. If we naively set ε = 0 {\displaystyle \varepsilon =0} , we would get the solution labelled "outer" below which does not model the boundary layer, for which x is close to zero. For more details that show how to obtain the uniformly valid approximation, see method of matched asymptotic expansions.
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