Asymptotic analysis

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In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are

  • In applied mathematics, asymptotic analysis is used to build numerical methods to approximate equation solutions.
  • in computer science in the analysis of algorithms, considering the performance of algorithms when applied to very large input datasets.
  • the behavior of physical systems when they are very large, an example being Statistical mechanics.
  • in accident analysis when identifying the causation of crash through count modeling with large number of crash counts in a given time and space.

The simplest example, when considering a function f(n), is when there is a need to describe its properties as Template:Mvar becomes very large. Thus, if f(n) = n2+3n, the term 3Template:Mvar becomes insignificant compared to Template:Mvar2, when Template:Mvar is very large. The function f(n) is said to be "asymptotically equivalent to n2 as Template:Mvar → ∞", and this is written symbolically as f(n) ~ n2.

Definition

Formally, given functions Template:Mvar and Template:Mvar of a natural number variable Template:Mvar, one defines a binary relation

if and only if (according to Erdelyi, 1956)

This relation is an equivalence relation on the set of functions of Template:Mvar. The equivalence class of Template:Mvar informally consists of all functions Template:Mvar which are approximately equal to Template:Mvar in a relative sense, in the limit.

Asymptotic expansion

An asymptotic expansion of a function f(x) is in practice an expression of that function in terms of a series, the partial sums of which do not necessarily converge, but such that taking any initial partial sum provides an asymptotic formula for Template:Mvar. The idea is that successive terms provide an increasingly accurate description of the order of growth of Template:Mvar. An example is Stirling's approximation.

In symbols, it means we have

but also

and

for each fixed k, while some limit is taken, usually with the requirement that gk+1 = o(gk), cf little o notation, which means the (gk) form an asymptotic scale.

The requirement that the successive sums improve the approximation may then be expressed as

In case the asymptotic expansion does not converge, for any particular value of the argument there will be a particular partial sum which provides the best approximation and adding additional terms will decrease the accuracy. However, this optimal partial sum will usually have more terms as the argument approaches the limit value.

Asymptotic expansions typically arise in the approximation of certain integrals (Laplace's method, saddle-point method, method of steepest descent) or in the approximation of probability distributions (Edgeworth series). The famous Feynman graphs in quantum field theory are another example of asymptotic expansions which often do not converge.

Use in applied mathematics

Asymptotic analysis is a key tool for exploring the ordinary and partial differential equations which arise in the mathematical modelling of real-world phenomena.[1] An illustrative example is the derivation of the boundary layer equations from the full Navier-Stokes equations governing fluid flow. In many cases, the asymptotic expansion is in power of a small parameter, Template:Mvar: in the boundary layer case, this is the nondimensional ratio of the boundary layer thickness to a typical lengthscale of the problem. Indeed, applications of asymptotic analysis in mathematical modelling often[1] centre around a nondimensional parameter which has been shown, or assumed, to be small through a consideration of the scales of the problem at hand.

Method of dominant balance

The method of dominant balance is used to determine the asymptotic behavior of solutions to an ODE without fully solving it. The process is iterative, in that the result obtained by performing the method once can be used as input when the method is repeated, to obtain as many terms in the asymptotic expansion as desired.[2]

The process goes as follows:

  • 1. Assume that the asymptotic behavior has the form
  • 2. Make an informed guess as to which terms in the ODE might be negligible in the limit of interest.
  • 3. Drop these terms and solve the resulting simpler ODE.
  • 4. Check that the solution is consistent with step 2. If this is the case, then one has the controlling factor of the asymptotic behavior; otherwise, one needs try dropping different terms in step 2, instead.
  • 5. Repeat the process to higher orders, relying on the above result as the leading term in the solution.

Example. For arbitrary constants Template:Mvar and Template:Mvar, consider

This differential equation cannot be solved exactly. However, it may be useful to know how the solutions behave for large Template:Mvar. Start by assuming as x → ∞; we do this with the benefit of hindsight, to make things quicker. Since we mostly care about the behavior of Template:Mvar in the large Template:Mvar limit, we change variables to Template:Mvar = exp(S(x)), and re-express the ODE in terms of S(x),

,

or

where we have used the product rule and chain rule to evaluate the derivatives of Template:Mvar.

Now suppose first that a solution to this ODE satisfies

as x → ∞, so that

as x → ∞. Obtain then the dominant asymptotic behaviour by setting

If satisfies the above asymptotic conditions, then the above assumption is consistent. The terms we dropped will indeed have been negligible with respect to the ones we kept.

is not a solution to the ODE for Template:Mvar, but it represents the dominant asymptotic behaviour, which is what we are interested in. Check that this choice for is consistent,

Everything is indeed consistent.

Thus the dominant asymptotic behaviour of a solution to our ODE has been found,

By convention, the full asymptotic series is written as

so to get at least the first term of this series we have to take a further step to see if there is a power of Template:Mvar out the front.

We proceed by introducing a new subleading dependent variable,

and then seek asymptotic solutions for C(x). Substituting into the above ODE for S(x) we find

Repeating the same process as before, we keep C' and (c-a)/x to find that

The leading asymptotic behaviour is then

See also

References

  1. 1.0 1.1 S. Howison, Practical Applied Mathematics, Cambridge University Press, Cambridge, 2005. ISBN 0-521-60369-2
  2. {{#invoke:citation/CS1|citation |CitationClass=book }}
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  • Asymptotic Expansions (Dover Books on Mathematics) by A. Erdelyi, 1956.