Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit

TL;DR

This paper introduces a self-similar approximation method to predict large-variable behavior of nonlinear problems using only small-variable series, demonstrated through physics examples.

Contribution

The paper presents a novel self-similar factor approximation technique that accurately extrapolates solutions from small to large variables in nonlinear problems.

Findings

Method accurately predicts large-variable behavior from small-variable series.

Numerical convergence of the approximants is demonstrated.

Applicable to rational, irrational, and transcendental functions.

Abstract

Complicated physical problems usually are solved by resorting to perturbation theory leading to solutions in the form of asymptotic series in powers of small parameters. However, finite, and even large values of the parameters often are of the main physical interest. A method is described for predicting the large-variable behavior of solutions to nonlinear problems from the knowledge of only their small-variable expansions. The method is based on self-similar approximation theory resulting in self-similar factor approximants. The latter can well approximate a large class of functions, rational, irrational, and transcendental. The method is illustrated by several examples from statistical and condensed matter physics, where the self-similar predictions can be compared with the available large-variable behavior. It is shown that the method allows for finding the behavior of solutions at large variables when knowing just a few terms of small-variable expansions. Numerical convergence of approximants is demonstrated.