Asymptotic expansion is one of the methods of approximating functions by their asymptotic expressions. The function g(x) is an asymptotic expression of the function f(x) for x → a , if
Hence it follows that
i.e., a relative error of replacing the function by its asymptotic expression for x → a tends to zero.
An asymptotic expansion is a special kind of an asymptotic expression, in which the function f(x) is approximated by partial sums of some convergent or divergent series
so that partial sums Sk = φ0(x) + φ1(x) + ... + φk(x) are the asymptotic expressions of the function f(x) and
i.e., each successive partial sum is the best asymptotic expression for the function f(x).
If the form of the function φi(x) is unknown beforehand, there exists a broad range of possibilities for choosing particular asymptotic expansions.
In the theory of fluid dynamics and heat mass transfer, and also in other branches of mechanics of continuous media the method of perturbation has found extensive application, in which approximate solutions of the problem, which is defined by a single or a system of (integro) differential equations and by the corresponding boundary conditions, are found through asymptotic expansions of dependent variables when one or several parameters of the problem (for instance, numbers M, Re, Pr, Sc etc., or their complexes) are small or large (in this latter case reciprocal values of the parameters are considered). Usually, it suffices to only define the first, or more rarely the first and the second approximations; the rest of the approximations serve for adjusting the first one. As a rule, the application of perturbation theory brings about a solution of more simple differential equations, of a smaller dimension or order, and also for different equations in partial derivatives, of more simple equations of another type (for instance, parabolic equations of a boundary layer approximation instead of Navier-Stokes elliptic equations in problems of dynamics of viscous liquid for large Reynolds numbers).
If the asymptotic expansion in terms of a small parameter is uniformly exact in the entire domain of definition of independent variables, a problem of regular perturbations holds; if it is nonuniformly exact, the problem is of singular perturbations. Among the methods of solution of the last problem the methods of deformed coordinates, of joint asymptotic expansions and of many scales are widely employed.
Nayfeh, (1973) Perturbation Methods. J. Wiley & Sons, New York. London, Sydney, Toronto.
Van Dyke, M. (1964) Perturbation Methods in Fluid Mechanics. Acad. Press, New York, London.
Cole, J. D. (1968) Perturbation Methods in Applied Mathematics. Blaisdell Pull. Comp., Toronto, London.