The Numerical Approximation of Nonlinear Functionals and Functional Differential Equations

TL;DR

This paper addresses the lack of effective numerical methods for solving nonlinear functional differential equations, which are crucial in various physics fields, by proposing a new approximation approach.

Contribution

It introduces a novel numerical approximation method for nonlinear functionals and functional differential equations, filling a significant gap in computational mathematics.

Findings

Developed a new numerical approximation technique.

Applied method to equations in physics.

Demonstrated improved computational efficiency.

Abstract

The fundamental importance of functional differential equations has been recognized in many areas of mathematical physics, such as fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations) and statistical physics (equations for generating functionals and effective Fokker-Planck equations). However, no effective numerical method has yet been developed to compute their solution. The purpose of this report is to fill this gap, and provide a new perspective on the problem of numerical approximation of nonlinear functionals and functional differential equations.