Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a ‘hyperasymptotic’ approximation. This adds a se...

In this article, by using Chebyshev’s polynomials and Chebyshev’s expansion, we obtain the best uniform polynomial approximation out of P2n to a class of rational functions of the form (ax2+c)-1 on any non symmetric interval [d,e]. Using the obtained approximation, we provide the best uniform polynomial approximation to a class of rational functions of the form (ax2+bx+c)-1 for both cases b2-4a...

We consider a one-dimensional defocusing Gross-Pitaevskii equation with a parabolic potential. Dark solitons oscillate near a center of the potential trap and their amplitude decays due to radiative losses (sound emission). We develop a systematic asymptotic multiscale expansion method in the limit when the potential trap is flat. The first-order approximation predicts a uniform frequency of os...

We characterize a regime of extreme (Jeans-like) instability, for strings evolving in cosmological backgrounds, in which the string's proper size grows asymptotically like the scale factor of the expanding Universe.We develop a new approximation scheme, based on the asymptotic proportionality of world-sheet and conformal times, for the systematic, quantitative description of such non-linear reg...

Let Wn := ∑n j=1 Zj be a sum of independent integer valued random variables. In this paper, we derive an asymptotic expansion for the probability IP[Wn ∈ A] of an arbitrary subset A ∈ Z+. Our approximation improves upon the classical expansions by including an explicit, uniform error estimate, involving only easily computable properties of the distributions of the Zj : an appropriate number of ...

Let (Bt; t ≥ 0) be a Brownian motion process starting from B0 = ν and define Xν(t) = ∫ t 0 Bs ds. For a≥ 0, set τa,ν := inf{t : Xν(t) = a} (with inf φ=∞). We study the conditional moments of τa,ν given τa,ν <∞. Using martingale methods, stopping-time arguments, as well as the method of dominant balance, we obtain, in particular, an asymptotic expansion for the conditional mean E(τa,ν | τa,ν <∞)...

We study the short time behavior of the early exercise boundary for American style put options in the Black–Scholes theory. We develop an asymptotic expansion which shows that the simple lower bound of Barles et al. is a more accurate approximation to the actual boundary than the more complex upper bound. Our expansion is obtained through iteration using a boundary integral equation. This integ...

It is shown that the presence of multiple time scales at a quantum critical point can lead to a breakdown of the loop expansion for critical exponents, since coefficients in the expansion diverge. Consequently, results obtained from finite-order perturbative renormalization-group treatments may not be an approximation in any sense to the true asymptotic critical behavior. This problem manifests...

The macroscopic description of a quantum oscillator with linear passive dissipation is formulated in terms of a master equation for the reduced density matrix. The procedure used is based on the asymptotic methods of nonlinear dynamics, which enables one to obtain an expression for the general term in the weak coupling expansion. For the special example of a charged oscillator interacting with ...