Singular perturbation

Singular perturbation theory

When we apply the steady-state approximation (SSA) in chemical kinetics, we typically argue that some of the intermediates are highly reactive, so that they are removed as fast as they are made. We then set the corresponding rates of change to zero. What we are saying is not that these rates are identically zero, of course, but that they are much smaller than the other rates of reaction. The st...

Geometric Singular Perturbation Theory

A Schrödinger singular perturbation problem

Consider the equation −ε∆uε + q(x)uε = f(uε) in R3, |u(∞)| < ∞, ε = const > 0. Under what assumptions on q(x) and f(u) can one prove that the solution uε exists and limε→0 uε = u(x), where u(x) solves the limiting problem q(x)u = f(u)? These are the questions discussed in the paper.

A nonlinear singular perturbation problem

Let F (uε) + ε(uε − w) = 0 (1) where F is a nonlinear operator in a Hilbert space H, w ∈ H is an element, and ε > 0 is a parameter. Assume that F (y) = 0, and F ′(y) is not a boundedly invertible operator. Sufficient conditions are given for the existence of the solution to (1) and for the convergence limε→0 ‖uε−y‖ = 0. An example of applications is considered. In this example F is a nonlinear ...

Applications of Singular Perturbation Methods in Queueing

A survey is presented describing the application of singular perturbation techniques to queueing systems. The goal is to compute performance measures by constructing approximate solutions to specific problems involving either the Kolmogorov forward or backward equation which contain a small parameter. These techniques are particularly useful on problems for which exact solutions are not availab...