Propagating front solutions in a time-fractional Fisher-KPP equation

TL;DR

This paper investigates the propagation speed of solutions in a time-fractional Fisher-KPP equation, introducing the concept of asymptotic traveling wave solutions to account for fractional derivatives.

Contribution

It proposes the notion of asymptotic traveling wave solutions for the fractional Fisher-KPP equation and analyzes their existence and properties using a monotone iteration method.

Findings

Asymptotic traveling wave solutions exist for the fractional Fisher-KPP equation.

Numerical simulations support the theoretical analysis of solution behavior.

The propagation speed is characterized through the asymptotic wave solutions.

Abstract

In this paper, we treat the Fisher-KPP equation with a Caputo-type time fractional derivative and discuss the propagation speed of the solution. The equation is a mathematical model that describes the processes of sub-diffusion, proliferation, and saturation. We first consider a traveling wave solution to study the propagation of the solution, but we cannot define it in the usual sense due to the time fractional derivative in the equation. We therefore assume that the solution asymptotically approaches a traveling wave solution, and the asymptotic traveling wave solution is formally introduced as a potential asymptotic form of the solution. The existence and the properties of the asymptotic traveling wave solution are discussed using a monotone iteration method. Finally, the behavior of the solution is analyzed by numerical simulations based on the result for asymptotic traveling wave solutions.