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Thus the leading-order behaviour of this equation at x=10 is that y increases cubically with x. The main behaviour of y may thus be investigated at any value of x. The leading-order behaviour is more complicated when more terms are leading-order. At x=2 there is a leading-order balance between the cubic and linear dependencies of y on x.
First-order approximation is the term scientists use for a slightly better answer. [3] Some simplifying assumptions are made, and when a number is needed, an answer with only one significant figure is often given ("the town has 4 × 10 3, or four thousand, residents"). In the case of a first-order approximation, at least one number given is exact.
Higher-order approximations will involve additional parameters. For weak curvature effects (small De), the Dean equations can be solved as a series expansion in De. The first correction to the leading-order axial Poiseuille flow is a pair of vortices in the cross-section carrying flow from the inside to the outside of the bend across the centre ...
The leading order term of the expansion is given by the linear noise approximation, in which the master equation is approximated by a Fokker–Planck equation with linear coefficients determined by the transition rates and stoichiometry of the system.
Comparison of Stirling's approximation with the factorial. In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of .
However, if the variation is very slow, then the WKBJ approximation may be used to derive a leading-order approximation to the solution. This gives rise to the theory of global modes, which was first developed by Philip Drazin in 1974. [3]
An approximation in the form of an asymptotic series is obtained in the transition layer(s) by treating that part of the domain as a separate perturbation problem. This approximation is called the inner solution, and the other is the outer solution, named for their relationship to the transition layer(s). The outer and inner solutions are then ...
Given approximations of from three distinct step sizes , /, and /, the exact relationship = () + = () + yields an approximate relationship (please note that the notation here may cause a bit of confusion, the two O appearing in the equation above only indicates the leading order step size behavior but their explicit forms are different and ...