Self-similar Differential Equations

TL;DR

This paper introduces self-similar differential equations (SSDEs), establishes their existence and uniqueness, and presents a novel class of equations inspired by similarity transformations, expanding differential equations theory.

Contribution

The paper defines SSDEs, proves their existence and uniqueness, and is the first to explore this new class of equations.

Findings

Existence and uniqueness of solutions for SSDEs

SSDEs are not ordinary differential equations

First formal study of self-similar differential equations

Abstract

Differential equations where the graph of some derivative of a function is composed of a finite number of similarity transformations of the graph of the function itself are defined. We call these self-similar differential equations (SSDEs) and prove existence and uniqueness of solution under certain conditions. While SSDEs are not ordinary differential equations, the technique for demonstrating existence and uniqueness of SSDEs parallels that for ODEs. This paper appears to be the first work on equations of this nature.