Fractional Reaction-diffusion Systems Research Articles

GROUND STATES FOR A FRACTIONAL REACTION-DIFFUSION SYSTEM

In this paper, we prove the existence of the ground state of a strongly indefinite fractional reaction-diffusion system based on the Non-Nehari method established by Tang-Chen-Lin-Yu [J. Differ. Equ., 2020(268), 4663-4690]. In particular, neither any monotonicity condition nor any Ambrosetti-Rabinowitz growth condition is required. To our knowledge, this is the first result about the ground states with the strongly indefinite case for fractional reaction-diffusion system.

Boundary Mittag-Leffler Stabilization of a Class of Time Fractional Order Nonlinear Reaction-Diffusion Systems

Boundary Mittag-Leffler Stabilization of a Class of Time Fractional Order Nonlinear Reaction-Diffusion Systems

A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition

<p style='text-indent:20px;'>This paper is devoted to the mathematical and numerical study of a new proposed model based on a fractional diffusion equation coupled with a nonlinear regularization of the Total Variation operator. This model is primarily intended to introduce a weak norm in the fidelity term, where this norm is considered more appropriate for capturing very oscillatory characteristics interpreted as a texture. Furthermore, our proposed model profits from the benefits of a variable exponent used to distinguish the features of the image. By using Faedo-Galerkin method, we prove the well-posedness (existence and uniqueness) of the weak solution for the proposed model. Based on the alternating direction implicit method of Peaceman-Rachford and the approximations of the Gr<inline-formula><tex-math id="M1">\begin{document}$ \ddot{u} $\end{document}</tex-math></inline-formula>nwald-Letnikov operators, we develop the numerical discretization of our fractional diffusion equation. Experimental results claim that our model provides high-quality results in cartoon-texture-edges decomposition and image denoising. In particular, our model can successfully reduce the staircase phenomenon during the image denoising. Furthermore, small details, texture and fine structures still maintained in the restored image. Finally, we compare our numerical results with the existing models in the literature.

Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems

Abstract The aim of this paper is to investigate a class of nonlinear stochastic reaction-diffusion systems involving fractional Laplacian in a bounded domain. First, the existence and uniqueness of weak solutions are proved by using Galërkin’s method. Second, the existence of optimal controls for the corresponding stochastic optimal control problem is obtained. Finally, several examples are provided to demonstrate the theoretical results.

Boundary feedback stabilization for a class of distributed-order fractional reaction diffusion systems

Boundary feedback stabilization for a class of distributed-order fractional reaction diffusion systems

Chaotic and spatiotemporal oscillations in fractional reaction-diffusion system

• Fractional derivative approximation in 2D. • Linear stability analysis for temporal system. • Alternating direction implicit finite difference method. • Numerical simulations. • Chaotic and spatiotemporal phenomenon. This paper focuses on the design and analysis of an efficient numerical method based on the novel implicit finite difference scheme for the solution of the dynamics of reaction-diffusion models. The work replaces the integer first order derivative in time with the Caputo fractional derivative operator. The dynamics of activator-inhibitor as encountered in chemistry, physics and engineering processes, and predator-prey models are two cases addresses in this study. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction-diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve cross-diffusion problem in two-dimensions. In the experimental results, a number of spatiotemporal and chaotic patterns that are related to Turing pattern are observed. It was discovered in the simulation experiments that the species predator-prey model distribute in almost same fashion, while that of the activator-inhibitor dynamics behaved differently regardless of the value of fractional order chosen.

Global existence and large time behavior of solutions of a time fractional reaction diffusion system

Abstract In this paper, it is proved that a time fractional reaction diffusion system with reaction terms of the Brusselator type admits a global solution by using the feedback method of F. Rothe [20]. Furthermore, some results on the large time behavior of the solutions are obtained. We give a positive answer to Problem 6 of the valuable paper of Gal and Warma [6].

High-dimensional spatial patterns in fractional reaction-diffusion system arising in biology

Abstract The concept of fractional derivative has been demonstrated to be successful when applied to model a range of physical and real life phenomena, be it in engineering and science related fields. It is a known fact that reaction-diffusion equation permits the use of different numerical methods in space and time. As a result, we introduce the Fourier spectral method for the discretization of space fractional derivative and adapt the modified version of the exponential time-integrator to advance in time in attempt to explore the dynamic richness of fractional reaction-diffusion equations in two and three dimensions. This approach gives a full diagonal representation of the fractional derivative operator and yields a better spectral convergence irrespective of the value of fractional order chosen in the experiment. Recommendations are made based on some amazing results which arise from the computational experiments. We intend to answer the question ’why is wildlife animals going into extinction in Africa?’ to a reasonable extent. We believe that the spatial patterns obtained in the simulation framework to mimic the ones found in wildlife would provide a measure and serves as a good alternative to an act of killing of wildlife animals for ornamental and decorative purposes, also would serve as a guild to textile industries on pattern formations.

Blow up of nonautonomous fractional reaction-diffusion systems

Blow up of nonautonomous fractional reaction-diffusion systems

Mathematical modelling of pattern formation in activator–inhibitor reaction–diffusion systems with anomalous diffusion

Auto-wave solutions in nonlinear time-fractional reaction–diffusion systems are investigated. It is shown that stability of steady-state solutions and their subsequent evolution are mainly determined by the eigenvalue spectrum of a linearized system and level of anomalous diffusion (orders of fractional derivatives). The results of linear stability analysis are confirmed by computer simulations. To illustrate the influence of anomalous diffusion on stability properties and possible dynamics in fractional reaction–diffusion systems, we generalized two classical activator–inhibitor nonlinear models: FitzHugh–Nagumo and Brusselator. Based on them a common picture of typical nonlinear solutions in nonlinear incommensurate time-fractional activator–inhibitor systems is presented.

Asymptotic stability conditions for autonomous time–fractional reaction–diffusion systems

This paper establishes conditions for the asymptotic stability of time–fractional reaction–diffusion systems. The stability of linear systems is investigated by means of the eigenfunction expansion of the Laplacian operator. Theoretical bounds are placed on the arguments of the infinity of eigenvalues belonging to the instant Jacobian matrix. Nonlinear systems are linearized by means of their Taylor series expansion. Numerical solutions of two realistic examples are presented to illustrate the theoretical findings.

Computational study of multi-species fractional reaction-diffusion system with ABC operator

Abstract In this paper, a competition model which describes the spatial interaction among three species in nonlinear fashion is considered. In the model, the standard time derivative is replaced with the Atangana-Baleanu fractional operator in the sense of Caputo. Linear stability analysis which serves as a guide in the choice of parameters when numerically simulating the full system is also examined. The existence and uniqueness of solutions are studied via a fixed point theorem. Different numerical approximation techniques are introduced. Numerical results presented in one and two dimensions revealed some spatiotemporal Turing patterns such as stripes and spots.

Modeling and simulation of nonlinear dynamical system in the frame of nonlocal and non-singular derivatives

This paper considers mathematical analysis and numerical treatment for fractional reaction-diffusion system. In the model, the first-order time derivatives are modelled with the fractional cases of both the Atangana-Baleanu and Caputo-Fabrizio derivatives whose formulations are based on the notable Mittag-Leffler kernel. The main system is examined for stability to ensure the right choice of parameters when numerically simulating the full model. The novel Adam-Bashforth numerical scheme is employed for the approximation of these operators. Applicability and suitability of the techniques introduced in this work is justified via the evolution of the species in one and two dimensions. The results obtained show that modelling with fractional derivative can give rise to some Turing patterns.

On a fractional reaction–diffusion system applied to image decomposition and restoration

In this paper, we propose a new model using a fractional reaction–diffusion system for image restoration and image decomposition which is motivated by work of S. Osher, A. Solé and L. Vese’s. Our model decomposes the degraded image into cartoon component belonging to a fractional Sobolev space and textured component belonging to a negative Hilbert space. We are interested in the existence and uniqueness of weak solutions of the system obtained from the model. To overcome the difficulties of the singularity in the system, we use a regularization method. We also give a numerical algorithm for solving the system, and present numerical experimental results to show the efficiency of our model.

Synchronization results for a class of fractional-order spatiotemporal partial differential systems based on fractional Lyapunov approach

In this paper, the problem of synchronization of a class of spatiotemporal fractional-order partial differential systems is studied. Subject to homogeneous Neumann boundary conditions and using fractional Lyapunov approach, nonlinear and linear control schemes have been proposed to synchronize coupled general fractional reaction–diffusion systems. As a numerical application, we investigate complete synchronization behaviors of coupled fractional Lengyel–Epstein systems.

Turing pattern in the fractional Gierer–Meinhardt model**Project supported by the National Natural Science Foundation of China (Grant Nos. 61573008 and 61703290) and Natural Science Foundation of Liaoning Province, China (Grant No. 20180550996).

It is well-known that reaction–diffusion systems are used to describe the pattern formation models. In this paper, we will investigate the pattern formation generated by the fractional reaction–diffusion systems. We first explore the mathematical mechanism of the pattern by applying the linear stability analysis for the fractional Gierer–Meinhardt system. Then, an efficient high-precision numerical scheme is used in the numerical simulation. The proposed method is based on an exponential time differencing Runge–Kutta method in temporal direction and a Fourier spectral method in spatial direction. This method has the advantages of high precision, better stability, and less storage. Numerical simulations show that the system control parameters and fractional order exponent have decisive influence on the generation of patterns. Our numerical results verify our theoretical results.

Mathematical analysis and numerical simulation of a fractional reaction-diffusion system with Holling-type III functional response

In recent years, many investigators have questioned the use of convectional diffusion equation to model many physical or real life situations. As a result, fractional space derivatives have been proposed to model anomalous diffusion or related processes, where a particle plume spreads at inconsistent rate with the classical Brownian motion model. By replacing the second derivative in the classical diffusion model with fractional derivative, results to enhance a process known as superdiffusion. A high-dimensional predator-prey reaction-diffusion system with Holling-type III functional response, where the usual second-order derivatives give place to a fractional derivative of order α with 1 < α ≤ 2. Analysis of the main equation guides in the correct choice of parameter values. We established the condition for local and global stabilities. We also show that the system undergoes a Hopf bifurcation subject to a small perturbation of the steady-state solution. The complexity of fractional derivative at some instances of order α for the superdiffusive scenario is demonstrated with some numerical experiments in one, two and three dimensions. The effectiveness of the numerical method is demonstrated through numerical simulations to confirm the theoretical results.

Mathematical analysis and numerical simulation of a fractional reaction-diffusion system with Holling-type III functional response

In recent years, many investigators have questioned the use of convectional diffusion equation to model many physical or real life situations. As a result, fractional space derivatives have been proposed to model anomalous diffusion or related processes, where a particle plume spreads at inconsistent rate with the classical Brownian motion model. By replacing the second derivative in the classical diffusion model with fractional derivative, results to enhance a process known as superdiffusion. A high-dimensional predator-prey reaction-diffusion system with Holling-type III functional response, where the usual second-order derivatives give place to a fractional derivative of order α with 1 < α ≤ 2. Analysis of the main equation guides in the correct choice of parameter values. We established the condition for local and global stabilities. We also show that the system undergoes a Hopf bifurcation subject to a small perturbation of the steady-state solution. The complexity of fractional derivative at some instances of order α for the superdiffusive scenario is demonstrated with some numerical experiments in one, two and three dimensions. The effectiveness of the numerical method is demonstrated through numerical simulations to confirm the theoretical results.

Event-triggered boundary feedback control for networked reaction-subdiffusion processes with input uncertainties

This paper is concerned with the event-triggered boundary feedback control problems for networked reaction-subdiffusion processes governed by time fractional reaction-diffusion systems with unknown time-varying input uncertainties over sensor/actuator networks. The event-triggered boundary state feedback controller is first designed and implemented via backstepping technique. Moreover, we realize that the availability of full-state measurements in many practical applications may be impossible due to the difficulties in measuring. To solve this limitation, we design an extended Luenberger observer that embeds within the networked sensor to estimate the whole states of the systems under consideration. Based on this, the boundary output feedback event-triggered implementation of the studied system in the context of sensor/actuator networks is then proposed. It is shown that both two kinds of event-triggered strategies could significantly asymptotically stabilize the estimation with the Zeno phenomenon being excluded. Two numerical illustrations are finally included.

Observer‐based output feedback control for a boundary controlled fractional reaction diffusion system with spatially‐varying diffusivity

This study is concerned with observer design and observer-based output feedback control for a fractional reaction diffusion (FRD) system with a spatially-varying (non-constant) diffusion coefficient by the backstepping method. The considered FRD system is endowed with only boundary measurable and actuation available. The contribution of this study is divided into three parts: first is the backstepping-based observer design for the FRD system with non-constant diffusivity, second is the output feedback controller generated by the integration of a separately backstepping-based feedback controller and the proposed observer to stabilise the FRD system with non-constant diffusivity, and the last is the Mittag–Leffler stability analysis of the observer error and the closed-loop FRD systems. Specifically, anti-collocated location of actuator and sensor is considered in the stabilisation problem of this system with Robin boundary condition at x = 0 and the boundary feedback controller for Dirichlet actuation at x = 1 . By designing an invertible coordinate transformation to convert the observer error system into a Mittag–Leffler stable target system, the observer gains are obtained. They are used to design the output feedback control law for stabilising the closed-loop system. Finally, a numerical example is shown to validate the effectiveness of the authors' proposed method.