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Mode Analysis on Onset of Turing Instability in Time-Fractional Reaction-Subdiffusion Equations by 2D Numerical Simulations

[+] Author and Article Information
Masataka Fukunaga

P. T. Lecturer, College of Engineering, Nihon UNiversity, (home) 1-2-35-405, Katahira, Aoba-ku, Sendai, Miyagi 980-0812, Japan
fukunaga@image.ocn.ne.jp

1Corresponding author.

ASME doi:10.1115/1.4043149 History: Received October 07, 2018; Revised March 04, 2019

Abstract

There are two types of time-fractional reaction-subdiffusion equations for two species. One of them generalizes the time derivative of species to frac- tional order, while in the other type the diffusion term is differentiated with respect to time of fractional order. In the latter equation, the Turing insta- bility appears as oscillation of concentration of species. In this paper, it is shown by the mode analysis that the critical point for the Turing instability is the standing oscillation of the concentrations of the species that does not decay nor increase with time. In special cases in which the fractional order is a rational number, the critical point is derived analytically by mode analysis of linearized equaitons. However, in most cases, the critical point is derived numerically by the linearized equations and 2D simulations. As a by-product of mode analysis, a method of checking the accuracy of numerical fractional reaction-subdiffusion equation is found. The solutions of the linearized equa- tion at the critical points are used to check accuracy of discretized model of 1D and 2D fractional reaction-diffusion equations.

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