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

Hiroshi Ishii

TL;DR

This paper studies front propagation in a time-fractional Fisher–KPP equation with Caputo derivative, addressing the challenge that classical traveling waves are not well-defined due to time nonlocality. It introduces the notion of an asymptotic traveling wave and proves existence for all speeds c ≥ c*α, where c*α = (2^{1/α}/√α)( f'(0)/(2−α) )^{(2−α)/(2α)} and the tail is controlled by the root λ1 of V(λ)=λ^2−(cλ)^α+f'(0). Using a monotone-iteration scheme with Green kernels, the authors construct upper and lower solutions and establish results for both c>c*α and c=c*α, linking the wave behavior to the linearized problem via λ1. Numerical experiments implementing a LX time-stepping scheme and finite differences corroborate the theoretical findings, showing front-like propagation with speeds close to c*α and illustrating how α modulates front speed and shape. The work provides a framework to analyze propagation in sub-diffusive, saturating reaction-diffusion systems and highlights open questions about minimal speeds and convergence to asymptotic waves in time-nonlocal settings.

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.

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

TL;DR

This paper studies front propagation in a time-fractional Fisher–KPP equation with Caputo derivative, addressing the challenge that classical traveling waves are not well-defined due to time nonlocality. It introduces the notion of an asymptotic traveling wave and proves existence for all speeds c ≥ c*α, where c*α = (2^{1/α}/√α)( f'(0)/(2−α) )^{(2−α)/(2α)} and the tail is controlled by the root λ1 of V(λ)=λ^2−(cλ)^α+f'(0). Using a monotone-iteration scheme with Green kernels, the authors construct upper and lower solutions and establish results for both c>c*α and c=c*α, linking the wave behavior to the linearized problem via λ1. Numerical experiments implementing a LX time-stepping scheme and finite differences corroborate the theoretical findings, showing front-like propagation with speeds close to c*α and illustrating how α modulates front speed and shape. The work provides a framework to analyze propagation in sub-diffusive, saturating reaction-diffusion systems and highlights open questions about minimal speeds and convergence to asymptotic waves in time-nonlocal settings.

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.
Paper Structure (10 sections, 14 theorems, 122 equations, 2 figures)

This paper contains 10 sections, 14 theorems, 122 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Existence of asymptotic traveling wave solutions
  4. Preliminaries
  5. Proof of Theorem \ref{['thm:main']} in the case $c>c^{*}_{\alpha}$
  6. Proof of Theorem \ref{['thm:main']} in the case $c=c^{*}_{\alpha}$
  7. Numerical experiments
  8. Numerical scheme and setting
  9. Numerical results
  10. Discussion

Key Result

Theorem 2.2

Let $\alpha\in(0,1)$ and $c^{*}_{\alpha}:= \dfrac{2^{1/\alpha}}{\sqrt{\alpha}} \left( \dfrac{f'(0)}{2-\alpha} \right)^{(2-\alpha)/2\alpha}$. Then, for any $c\ge c^{*}_{\alpha}$ there exists an asymptotic traveling wave solution of eq:main with an increasing function $\phi$ satisfying boundary. Furth where $\lambda_1$ is the minimal positive root of $\lambda^2 - (c\lambda)^{\alpha}+f'(0)=0$.

Figures (2)

  • Figure 1: Spatio-temporal plots (top panels) and profiles of the numerical solutions (bottom panels). The parameters used are $l=300,\ l_0 = 295,\ \omega=0.01$, and $x_0 = 75$. The numerical simulations correspond to $\alpha=0.3$ (left), $\alpha=0.6$ (middle), and $\alpha=0.9$ (right). The solution profiles are shown at $T=60.04$ (left), $T=84.48$ (middle), and $T=106.05$ (right). See Subsection \ref{['subsec:num']} for how to determine $T$.
  • Figure 2: (a) Comparison of $c^{*}_{\alpha}$ (red line) and the propagation speed of the numerical solution (dashed line). The parameters are $l=300,\ l_0=295,\ \omega=0.01$, and $x_0 = 75$. The blue dashed line is constructed by connecting straight line segments computed for $\alpha=0.1 + 0.025\times k\ (0\le k \le 36)$. (b) Relative errors.

Theorems & Definitions (30)

  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 20 more