Approximate Solutions of Functional Equations

TL;DR

This paper develops methods for constructing approximate solutions to functional evolution equations using series and conjugation techniques, with analytical and numerical illustrations demonstrating improved accuracy in specific examples.

Contribution

It introduces combined series and conjugation methods for better approximate solutions to functional equations, with error estimates and practical illustrations.

Findings

Conjugation significantly improves approximation accuracy.

Methods effectively applied to functions like x/(1-x), sin x, and λx(1-x).

Numerical and analytical results confirm method effectiveness.

Abstract

Approximate solutions to functional evolution equations are constructed through a combination of series and conjugation methods, and relative errors are estimated. The methods are illustrated, both analytically and numerically, by construction of approximate continuous functional iterates for x/(1-x), sin x, and {\lambda}x(1-x). Simple functional conjugation by these functions, and their inverses, substantially improves the numerical accuracy of formal series approximations for their continuous iterates.