• Explorations of pattern formation in fractional partial differential equations
• This is the implementation of paper by Yin et.al. "Pattern formation through temporal fractional derivatives"
• May. 2, 2021 ~ Aug. 1, 2021

^∂ηu/_∂t = Δu + u(1-u) - ^uv/_u+α
^∂ηv/_∂t = dΔv - γ v + ^βuv/_u+α

If η = 1, the system degenerates into the temporal first-derivative system, which represents the instantaneous behaviors of the prey and predator.
The parameters are taken as α = 0.175, β = 0.95, γ = 0.5, η = 0.8, d = 20. ・・・ (a)

• Forward Euler method in time and Central differences in space (FTCS)
• Zero-flux boundary condition, 1D

The parameters are taken as (a).

If we change diffusion coefficient "d" into 0.1, we can observe the fluctuation in time and the small changes in space.
If "d" and "h" are properly chosen, even first-order derivatives can form patterns.

• It can form steadily spatial patterns even though its first-derivative counterpart can't exhibit any steady pattern.
• Approximation of the Caputo's derivative by the Grunwald-Letnikov one in time and Central differences in space
• Zero-flux boundary condition, 1D

The differences between 1, 2 with the same parameters implies that the fractional derivative can product steady-state spatial patterns and induce the Turing instability.

[1] Yin, Hongwei, and Xiaoqing Wen. "Pattern formation through temporal fractional derivatives." Scientific reports 8.1 (2018): 1-9.
[2] Ciesielski, Mariusz, and Jacek Leszczynski. "Numerical simulations of anomalous diffusion." arXiv preprint math-ph/0309007 (2003).