A new kind of functional differential equations

TL;DR

This paper introduces a novel class of functional differential equations, exploring their properties, solutions, and potential applications in mathematics and physics, highlighting their advantages over traditional equations.

Contribution

The paper presents a new type of functional differential equations, proving global existence of solutions and analyzing their unique properties, expanding the toolkit for mathematical and physical problem-solving.

Findings

Proved global existence of smooth solutions for specific examples

Identified unique properties of the new functional differential equations

Demonstrated potential applications in differential geometry and physics

Abstract

In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and PDEs. For some typical examples, we prove the global existence of smooth solutions, analyze some interesting properties enjoyed by these solutions, and illustrate the differences between this new class of equations and the traditional ones. This kind of functional differential equations is a new and powerful tool to study some problems arising from both mathematics and physics, more applications in particular to differential geometry and fundamental physics can be expected.