A Novel Solution for the General Diffusion

TL;DR

This paper introduces a new method to solve the Fisher-KPP reaction-diffusion equation, revealing that the solution becomes irregular over time and challenges the traditional traveling wave approximation used in such models.

Contribution

A novel approach for solving nonlinear PDEs applied to Fisher-KPP, showing solutions become irregular and invalidating common traveling wave assumptions.

Findings

Solution collapses to irregular form for all t > 0

Counterintuitive solution behavior confirmed by analysis

Traveling wave approximation is invalidated by the new solution

Abstract

The Fisher-KPP equation is a reaction-diffusion equation originally proposed by Fisher to represent allele propagation in genetic hosts or population. It was also proposed by Kolmogorov for more general applications. A novel method for solving nonlinear partial differential equations is applied to produce a unique, approximate solution for the Fisher-KPP equation. Analysis proves the solution is counterintuitive. Although still satisfying the maximum principle, time dependence collapses for all time greater than zero, therefore, the solution is highly irregular and not smooth, invalidating the traveling wave approximation so often employed.