# On the summation of divergent, truncated, and underspecified power series via asymptotic approximants

## TL;DR

This paper introduces a method called asymptotic approximants to accurately sum divergent or incomplete power series solutions by leveraging known asymptotic behavior, demonstrated on nonlinear physics problems.

## Contribution

The paper formalizes and provides an algorithm for constructing asymptotic approximants that bridge asymptotic regions, enabling accurate solutions from limited series data.

## Key findings

- Successfully constructed approximants for three nonlinear physics problems.
- Predicted unknown coefficients and properties with high accuracy.
- Provided new benchmark values for the Sakiadis boundary layer.

## Abstract

A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series solution or some truncation of the power series solution be available and that some asymptotic behavior of the solution is known away from the series' expansion point. Here, we formalize the method of asymptotic approximants that has found recent success in its application to thermodynamic virial series where only a few to (at most) a dozen series coefficients are typically known. We demonstrate how asymptotic approximants may be constructed using simple recurrence relations, obtained through the use of a few known rules of series manipulation. The result is an approximant that bridges two asymptotic regions of the unknown exact solution, while maintaining accuracy in-between. A general algorithm is provided to construct such approximants. To demonstrate the versatility of the method, approximants are constructed for three nonlinear problems relevant to mathematical physics: the Sakiadis boundary layer, the Blasius boundary layer, and the Flierl-Petviashvili monopole. The power series solution to each of these problems is underspecified since, in the absence of numerical simulation, one lower-order coefficient is not known; consequently, higher-order coefficients that depend recursively on this coefficient are also unknown. The constructed approximants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approximants lead to new benchmark values for the Sakiadis boundary layer and agree with recent numerical values for properties of the Blasius boundary layer and Flierl-Petviashvili monopole.