Background and Motivation
Power series arise in virtually all applications of mathematical physics. However, limitations generally inherent to power series solutions often inhibit their direct use. For instance, a Taylor series representation of an unknown function may not converge, as it may have a finite radius of convergence arising from singularities (often complex) in the function it represents. Even when singularities are not a concern, higher-order terms of the series may be exceedingly difficult to compute, which is especially problematic if the series converges slowly. While “re-summation” methods such as Pade Approximants typically lead to an implementation improvement compared with the original series, global accuracy is not always guaranteed and the problem becomes one of choosing the ‘ best’ re-summation technique.
Our goal is to identify relevant problems that can be solved using an asymptotic approximant: a function whose Taylor expansion (about $x=a$) matches the exact Taylor expansion to some order and whose limit as $x\to b$ matches some known asymptotic behavior. Divergent, truncated, and/or slowly converging series can be replaced by asymptotic approximants if some behavior is known away from the series expansion point. Nonlinear ODEs that require the "shooting method" are well suited for asymptotic approximants, applied to the (typically divergent) power series solution to the ODE. So far, we have successfully applied asymptotic approximants to find closed-form analytic solutions to problems in thermodynamics, fluid mechanics, and astrophysics.
Below are 4 examples of how asymptotic approximants (denoted A in the figures) can be used to bridge the behavior between two physical regions. In all figures, symbols represent numerical solutions. top left: 2nd, 4th, and 6th order approximants, whose zero-density expansion matches the square-well virial series (V) to 2nd, 4th, and 6th order, while also limiting to non-classical scaling behavior near the thermodynamic critical point (Barlow et al, JCP, 2015). top right: Approximant matches known weak (W)- and strong (S)-field limits for the bending angle of a Kerr black hole (Barlow, Weinstein, & Faber, CQG, 2017). bottom left: Approximant bridges the series solution (S) of the Blasius ODE (describing boundary layer flow over a plate) with its far-field behavior (Barlow et. al., QJMAM, 2017). bottom right: Approximant bridges the series solution of the Flierl-Petviashvilli ODE (describing vortex solitons like Jupiter's red spot) with the far field behavior (Barlow et. al., QJMAM, 2017).