Self-similar factor approximants for evolution equations and boundary-value problems

TL;DR

The paper introduces a simple, general method using self-similar factor approximants to solve evolution equations and boundary-value problems with high accuracy, often reconstructing exact solutions from asymptotic series.

Contribution

It presents a novel, straightforward two-step approach that improves upon perturbation theory for solving complex differential equations in physical applications.

Findings

Provides highly accurate approximate solutions

Can reconstruct exact solutions from asymptotic series

Applicable even when perturbation parameters are large

Abstract

The method of self-similar factor approximants is shown to be very convenient for solving different evolution equations and boundary-value problems typical of physical applications. The method is general and simple, being a straightforward two-step procedure. First, the solution to an equation is represented as an asymptotic series in powers of a variable. Second, the series are summed by means of the self-similar factor approximants. The obtained expressions provide highly accurate approximate solutions to the considered equations. In some cases, it is even possible to reconstruct exact solutions for the whole region of variables, starting from asymptotic series for small variables. This can become possible even when the solution is a transcendental function. The method is shown to be more simple and accurate than different variants of perturbation theory with respect to small parameters, being applicable even when these parameters are large. The generality and accuracy of the method are illustrated by a number of evolution equations as well as boundary value problems.