Asymptotic analysis

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

In applied mathematics, asymptotic analysis is used to build numerical methods to approximate equation solutions.
In mathematical statistics and probability theory, asymptotics are used in analysis of long-run or large-sample behaviour of random variables and estimators.
in computer science in the analysis of algorithms, considering the performance of algorithms.
the behavior of physical systems, 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.

A simple illustration, when considering a function $f (n)$ , is when there is a need to describe its properties as $n$ becomes very large. Thus, if $f (n) = n 2 +3 n$ , the term 3 $n$ becomes insignificant compared to $n$ ², when $n$ is very large. The function $f (n)$ is said to be "asymptotically equivalent to $n 2$ as $n$ → ∞", and this is written symbolically as $f (n) ~ n 2$ .

Definition

Formally, given functions $f$ and $g$ of a natural number variable $n$ , one defines a binary relation

f\sim g\quad ({\text{as }}n\to \infty )

if and only if (according to Erdelyi, 1956)

\lim _{n\to \infty }{\frac {f(n)}{g(n)}}=1~.

This relation is an equivalence relation on the set of functions of $n$ . The equivalence class of $f$ informally consists of all functions $g$ which are approximately equal to $f$ in a relative sense, in the limit.

Properties

If $f\sim g$ , then

$f^{r}\sim g^{r}$ for any real r, and

$\log(f)\sim \log(g)$ .

If $f\sim g$ and $a\sim b$ , then

$f\times a\sim g\times b$ , and

$f/a\sim g/b$ .

This allows asymptotically equivalent functions to be freely exchanged in many algebraic expressions.

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 $f$ . The idea is that successive terms provide an increasingly accurate description of the order of growth of $f$ . An example is Stirling's approximation.

In symbols, it means we have

f\sim g_{1}

but also

f-g_{1}\sim g_{2}

and

f-g_{1}-\cdots -g_{k-1}\sim g_{k}

for each fixed $k$ .

In view of the definition of the $\sim$ symbol, the last equation means

f-(g_{1}+\cdots +g_{k})=o(g_{k})

in the little o notation, i.e.,

f-(g_{1}+\cdots +g_{k})

is much smaller than

g_{k}

.

The relation

f-g_{1}-\cdots -g_{k-1}\sim g_{k}

takes its full meaning if

\forall k,g_{k+1}=o(g_{k})

,

which means the $g_{k}$ form an asymptotic scale.

In that case, some authors may abusively write

f\sim g_{1}+\cdots +g_{k}

to denote the statement

f-(g_{1}+\cdots +g_{k})=o(g_{k})\,.

One should however be careful that this is not a standard use of the $\sim$ symbol, and that it does not correspond to the definition given in § Definition.

In the present situation, this relation $g_{k}=o(g_{k-1})$ actually follows from combining steps $k$ and $(k -1)$ , by subtracting $f-g_{1}-\cdots -g_{k-2}=g_{k-1}+o(g_{k-1})$ from $f-g_{1}-\cdots -g_{k-2}-g_{k-1}=g_{k}+o(g_{k})$ one gets

g_{k}+o(g_{k})=o(g_{k-1})\,,

i.e., $g_{k}=o(g_{k-1})$ .

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, $ε$ : 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:

Assume that the asymptotic behavior has the form
$y(x)\sim e^{S(x)}~.$
Make an informed guess as to which terms in the ODE might be negligible in the limit of interest.
Drop these terms and solve the resulting simpler ODE.
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.
Repeat the process to higher orders, relying on the above result as the leading term in the solution.

Example. For arbitrary constants $c$ and $a$ , consider

xy''+(c-x)y'-ay=0~.

This differential equation cannot be solved exactly. However, it is useful to consider how the solutions behave for large $x$ : it turns out that $y$ behaves like $e^{x}\,$ as x → ∞ .

More rigorously, we will have $\log(y)\sim {x}$ , not $y\sim e^{x}$ . Since we are interested in the behavior of $y$ in the large $x$ limit, we change variables to $y$ = exp(S(x)), and re-express the ODE in terms of S(x),

xS''+xS'^{2}+(c-x)S'-a=0\,

,

or

S''+S'^{2}+\left({\frac {c}{x}}-1\right)S'-{\frac {a}{x}}=0\,

where we have used the product rule and chain rule to evaluate the derivatives of $y$ .

Now suppose first that a solution to this ODE satisfies

S'^{2}\sim S'~,

as x → ∞, so that

S'',~{\frac {c}{x}}S',~{\frac {a}{x}}=o(S'^{2}),~o(S')\,

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

S_{0}'^{2}=S_{0}'~.

If $S_{0}$ satisfies the above asymptotic conditions, then the above assumption is consistent. The terms we dropped will have been negligible with respect to the ones we kept.

$S_{0}$ is not a solution to the ODE for $S$ , but it represents the dominant asymptotic behavior, which is what we are interested in. Check that this choice for $S_{0}$ is consistent,

{\begin{aligned}S_{0}'&=1\\S_{0}'^{2}&=1\\S_{0}''&=0=o(S_{0}')\\{\frac {c}{x}}S_{0}'&={\frac {c}{x}}=o(S_{0}')\\{\frac {a}{x}}&=o(S_{0}')\end{aligned}}

Everything is indeed consistent.

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

{\begin{aligned}S_{0}&\sim x\\\log(y)&\sim x~.\end{aligned}}

By convention, the full asymptotic series is written as

y\sim Ax^{p}e^{\lambda x^{r}}\left(1+{\frac {u_{1}}{x}}+{\frac {u_{2}}{x^{2}}}\cdots +{\frac {u_{k}}{x^{k}}}+o\left({\frac {1}{x^{k}}}\right)\right)~,

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 $x$ out the front.

Proceed by introducing a new subleading dependent variable,

S(x)\equiv S_{0}(x)+C(x)\,

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

C''+C'^{2}+C'+{\frac {c}{x}}C'+{\frac {c-a}{x}}=0~.

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

C_{0}=\log x^{a-c}~.

The leading asymptotic behaviour is then

y\sim x^{a-c}e^{x}~.

References

^ ^a ^b S. Howison, Practical Applied Mathematics, Cambridge University Press, Cambridge, 2005. ISBN 0-521-60369-2
^ Bender, C.M.; Orszag, S.A. (1999). Advanced mathematical methods for scientists and engineers. Springer. pp. 549–568. ISBN 0-387-98931-5.

Boyd, John P. (March 1999). "The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series". Acta Applicandae Mathematicae. 56 (1): 1–98. doi:10.1023/A:1006145903624.
Asymptotic Expansions (Dover Books on Mathematics) by A. Erdelyi, 1956.

[Howison-1] S. Howison, Practical Applied Mathematics, Cambridge University Press, Cambridge, 2005. ISBN 0-521-60369-2

[2] Bender, C.M.; Orszag, S.A. (1999). Advanced mathematical methods for scientists and engineers. Springer. pp. 549–568. ISBN 0-387-98931-5.

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