Mathematical modeling of time fractional reaction–diffusion systems

Abstract

We study a fractional reaction-diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. The increase of the value of fractional derivative index leads to the time periodic solutions. The domains of existing periodic solutions for different parameters of the problem are obtained. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. For the fractional reaction-diffusion systems it is established that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary conditions. The characteristic features of these solutions consist of the transformation of the steady-state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system.

Mathematical Modelling of Subdiffusion-reaction Systems

“…It has been found that in the case of the Seki-Lindenberg model, (2.1), the oscillatory mode of a short-wave instability disappears. Note, that a short-wave oscillatory instability has been revealed for that model outside the region of its physical relevance for subdiffusion-limited reactions, when γ > γ 0 > 1 [51], [53], [57], [58].…”

Fractional heat conduction with heat absorption in a sphere under Dirichlet boundary condition

“…In the last few decades, differential equations with derivatives of non-integer order attract the attention of the researchers as such equations provide a very suitable tool for description of many important phenomena in physics, geophysics, chemistry, biology, engineering and solid mechanics (see, for example, Gafiychuk et al 2008;Herrmann 2011;Magin 2006;Mainardi 2010;Povstenko 2015a;Sabatier et al 2007;Tarasov 2010;Tenreiro Machado 2011;Uchaikin 2013).…”

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

“…While this does exclude certain applications, many interesting and relevant models can still be considered such as fractional phase-field models [12,14,15] or the fractional FokkerPlanck equation [48]. They also appear in many applications such as the anomalous diffusion [52], pattern formation using fractional derivatives [28], and also the simulation of fractional phase-field equations such as the Allen-Cahn equation [15]. Additionally, the low-rank approximation presented in this paper also applies in the case where the spatial domain is more complicated.…”