More precisely, the solution cannot be uniformly approximated by an asymptotic expansion 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.
The term "singular perturbation" was coined in the 1940s by Kurt Otto Friedrichs and Wolfgang R.
[1] 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
This corresponds to taking only the first term of the expansion, yielding an approximation that converges, perhaps slowly, to the true solution as
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.
Singular perturbation theory is a rich and ongoing area of exploration for mathematicians, physicists, and other researchers.
The numerical methods for solving singular perturbation problems are also very popular.
[2] For books on singular perturbation in ODE and PDE's, see for example Holmes, Introduction to Perturbation Methods,[3] Hinch, Perturbation methods[4] or Bender and Orszag, Advanced Mathematical Methods for Scientists and Engineers.
[5] 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.
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.
, 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.
An electrically driven robot manipulator can have slower mechanical dynamics and faster electrical dynamics, thus exhibiting two time scales.
In such cases, we can divide the system into two subsystems, one corresponding to faster dynamics and other corresponding to slower dynamics, and then design controllers for each one of them separately.
Through a singular perturbation technique, we can make these two subsystems independent of each other, thereby simplifying the control problem.
A theorem due to Tikhonov[6] states that, with the correct conditions on the system, it will initially and very quickly approximate the solution to the equations on some interval of time and that, as
decreases toward zero, the system will approach the solution more closely in that same interval.
Thus the fluid exhibits multiple spatial scales.
Reaction–diffusion systems in which one reagent diffuses much more slowly than another can form spatial patterns marked by areas where a reagent exists, and areas where it does not, with sharp transitions between them.
tend to zero, in that limit one of the roots escapes to infinity.
To prevent this root from becoming invisible to the perturbative analysis, we must rescale
will be chosen such that we rescale just fast enough so that the root is at a finite value of
This point where the highest order term will no longer vanish in the limit
to zero by becoming equally dominant to another term, is called significant degeneration; this yields the correct rescaling to make the remaining root visible.
are the two roots that we've found above that collapse to zero in the limit of an infinite rescaling.