Time-fractional Reaction Diffusion Systems Research Articles

A meshless approach based on fractional interpolation theory and improved neural network bases for solving non-smooth solution of 2D fractional reaction–diffusion equation with distributed order

The primary objective of this research paper is to present a novel and effective meshless numerical approach for solving the 2D time fractional reaction diffusion system with distributed order on an arbitrary domain. Gauss–Legendre quadrature formula is applied to discretize distributed-order derivative integral. We establish the piecewise parabolic fractional interpolation theory and with its assistance, the proposed approach can proficiently solve the non-smooth solutions of the equations and more accurately approximate the Caputo fractional derivative. This meshless method based on improved neural network bases combines the high accuracy approximation advantage and strong express ability of neural network to construct the basis functions set on arbitrary domains, which significantly reduces a computational consumption. The bases constructed based on the neural network allow the selection of 12 bases numbers to achieve an approximation equivalent to that of 1000 ordinary bases. The theoretical analyses of error and convergence order for the meshless approach are carried out. Numerical examples are implemented to validate the high precision and capability of the meshless numerical approach.

Stabilization of coupled delayed nonlinear time fractional reaction diffusion systems using sampled‐in‐space sensing and actuation

AbstractThis paper is considered with the asymptotic stabilization of coupled delayed nonlinear time fractional reaction diffusion systems (FRDSs) governed by fractional parabolic partial differential equations (PDEs) with space‐dependent coefficients under sampled‐data in space control. It is assumed that state measurements can be averaged measurements (AMs) or point measurements (PMs), and a finite number of sensing and actuation devices are located in a spaced manner along the spatial domain of the interest. With the proposed sampled‐data in space controller, the closed‐loop stability is obtained. Tuning rules of system parameters and control parameters are derived using the fractional Halanay's inequality and the fractional Lyapunov method. Subsequently, the dual problem of observer design is formulated. Fractional examples are used to valid the theoretical result. Discussions on the extension of sampled‐data boundary feedback stabilization are provided finally.

Solving Time-Fractional reaction–diffusion systems through a tensor-based parallel algorithm

Machine Learning (ML) approach is a discussed research topic because of its benefit in several research fields. The most important issues in the training process of ML are accuracy and speed: a suitable mathematical model is critical and a fast data processing is mandatory. Fractional Calculus is involved in a large number of important applications and, recently, many ML algorithms, in order to improve accuracy of results when performing training in solving optimization problems, are based on decision and control performed by means of time-fractional models to better understand complex systems. However, the high computational cost, which characterizes the numerical solution, of this approach might be a problem for large scale Machine Learning systems. High Performance Computing (HPC) is the way of addressing the need of real time computation. In fact, through tensor-based parallel strategies designed for modern parallel architectures, Fractional Calculus tools are very helpful for the ML training step. In this contest, we consider a time-fractional diffusion system and, after introducing a suitable modification of a numerical model to solve it, we propose a related and novel parallel implementation on GPUs (Graphics Processing Units). Experiments show the gain of performance in terms of execution time and accuracy of our parallel implementation.

An inverse problem of identifying two coefficients in a time-fractional reaction diffusion system

In this paper, we aim to study an inverse problem for determining two time-independent coefficients in a fractional diffusion system from the final measurements. First, we prove the well-posedness of the state problem, and then we show some regularity results for the solution of the direct system using the Mittag-Leffler function. Then, we reformulate our inverse problem into an optimal control one. Afterwards, we establish the existence of the minimizer and prove the stability estimate for two coefficients with respect to the final data. The descent method is proposed as a numerical one based on the gradient calculus via the adjoint state and we compare it with the conjugate gradient method. Finally, we will present some numerical tests that shows the efficiency of the proposed methods.

Asymptotic stabilisation of coupled delayed time fractional reaction diffusion systems with boundary input disturbances via backstepping sliding-mode control

In this paper we study the asymptotic stabilisation for coupled time fractional reaction diffusion (FRD) systems with time varying delays and input disturbances by backstepping-based boundary sliding-mode control. Here, the spatially varying diffusion coefficients can be same or distinct. To stabilise the system, we first use an invertible backstepping transformation to convert an original dynamics into a target dynamics with new manipulable inputs and perturbations. Then, we employ the sliding-mode algorithm to design this discontinuous controller to suppress disturbances. In this case, we obtain the combined backstepping/sliding-mode controller of the original dynamics. By using the fractional Halanay's inequality, the delay-independent stability condition is obtained for the asymptotic stabilisation of the delayed target system and therefore of the delayed original system under the combined controller. Fractional numerical examples are given to verify the efficiency of the proposed synthesis.

Observer design for time fractional reaction–diffusion systems with spatially varying coefficients and weighted spatial averages measurement

In this paper, we consider an observer design problem of a time fractional reaction–diffusion system with space-dependent coefficients and measurement of weighted spatial averages. Different from the boundary measurement, here, the output is a weighted spatial average of the state over the entire spatial domain. In order to apply the backstepping approach to obtain the observer gains, we first use a conversion of weighting functions to transform the error system into an intermediate system. With a change of variable, this intermediate system is then converted into the final target system with boundary measurement where backstepping is easily applicable. The resultant observer realises Mittag-Leffler convergence of the error system. Finally, numerical simulations are provided to check the effectiveness of the proposed observer.

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].

Trajectory planning for semilinear time-fractional reaction-diffusion systems under Robin boundary conditions

This paper describes how to design flatness-based distributed feedforward controllers for the solution of the trajectory planning problem of semilinear time-fractional reaction-diffusion systems (TFRDSs), where the first-order time derivative of conventional reaction-diffusion system is extended to a Caputo fractional derivative of order α ϵ (0,1]. To this end, an implicit system variable and control input parametrization is determined based on the spectral property of system operator with respect to a basic output and its fractional-order derivatives. The convergence of the parametrizations is guaranteed by restricting the basic output to some certain Gevrey classes. With these, we propose two approaches on solving the trajectory planning problem within a prescribed finite time interval. A simulation example is finally included to illustrate our results.

Mathematical Analysis and Numerical Simulation of Chaotic Noninteger Order Differential Systems with Riemann-Liouville Derivative

This article deals with the design, analysis, and implementation of a robust numerical scheme when applied to time-fractional reaction-diffusion system. Stability analysis and numerical treatment of chaotic fractional differential system in Riemann-Liouville sense are considered in this article. Simulation results show that chaotic phenomena can only occur if the reaction or local dynamics of such system is coupled or nonlinear in nature. Illustrative examples that are still of current and recurring interest to economists, engineers, mathematicians and physicists are chosen, to describe the points and queries that may arise. Numerical results presented agree with the theoretical findings.

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.

On the complete synchronization of a time-fractional reaction–diffusion system with the Newton–Leipnik nonlinearity

ABSTRACT In this paper, we consider a time-fractional reaction–diffusion system with the same nonlinearities of the Newton–Leipnik chaotic system. Through analytical tools and numerical results, we derive sufficient conditions for the asymptotic stability of the proposed model and show the existence of chaos. We also propose a nonlinear synchronization controller for a pair of systems and establish the local and global asymptotic convergence of the trajectories by means of fractional stability theory and the Lyapunov method.

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.

Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point

Abstract In this article, we study complex spatio-temporal solutions in nonlinear time-fractional reaction-diffusion systems. The main attention is paid to nonlinear dynamics near a bifurcation point. Despite the fact that the homogeneous state is stable at the parameters lower than bifurcation ones, a variety of complex solutions can also form in the subcritical domain. As an example, we consider a generalized fractional FitzHugh-Nagumo model. Depending on the given standard bifurcation parameters and the order of fractional derivative, the new types of steady auto-wave solutions in such systems have revealed. By computer simulation, it is shown that fractional reaction-diffusion possess much more complex nonlinear dynamics than their integer counterparts even at a subcritical bifurcation.

Global existence and asymptotic behavior for a time fractional reaction–diffusion system

This paper is concerned with the existence of global in time solutions of a time fractional reaction–diffusion system with time fractional derivatives. Furthermore, the large time behavior of bounded solutions is investigated. Our method of proof relies on a maximal regularity result for fractional linear reaction–diffusion equations that has been derived by Bajlekova (2001).

PATTERN FORMATION IN FRACTIONAL REACTION–DIFFUSION SYSTEMS WITH MULTIPLE HOMOGENEOUS STATES

This paper is devoted to the investigation of self-organization phenomena in time-fractional reaction–diffusion systems with multiple homogeneous states. It is shown that the fractional reaction–diffusion systems possess some new properties compared to the systems with derivatives of integer orders. In particular, some complex spatio-temporal solutions that cannot be found in the standard reaction–diffusion systems are identified. The simulation results are presented for the case of a incommensurate time-fractional reaction–diffusion system with a cubic nonlinearity.

Different Types of Instabilities and Complex Dynamics in Reaction-Diffusion Systems With Fractional Derivatives

In this article we analyze conditions for different types of instabilities and complex dynamics that occur in nonlinear two-component fractional reaction-diffusion systems. It is shown that the stability of steady state solutions and their evolution are mainly determined by the eigenvalue spectrum of a linearized system and the fractional derivative order. The results of the linear stability analysis are confirmed by computer simulations of the FitzHugh-Nahumo-like model. On the basis of this model, it is demonstrated that the conditions of instability and the pattern formation dynamics in fractional activator- inhibitor systems are different from the standard ones. As a result, a richer and a more complicated spatiotemporal dynamics takes place in fractional reaction-diffusion systems. A common picture of nonlinear solutions in time-fractional reaction-diffusion systems and illustrative examples are presented. The results obtained in the article for homogeneous perturbation have also been of interest for dynamical systems described by fractional ordinary differential equations.

Complex nonlinear dynamics in subdiffusive activator–inhibitor systems

In this article we analyze the linear stability of nonlinear time-fractional reaction–diffusion systems. As an example, the reaction–subdiffusion model with cubic nonlinearity is considered. By linear stability analysis and computer simulation, it was shown that fractional derivative orders can change substantially an eigenvalue spectrum and significantly enrich nonlinear system dynamics. A overall picture of nonlinear solutions in subdiffusive reaction–diffusion systems is presented.

Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems

In this article we investigate possible scenarios of pattern formations in reaction-diffusion systems with time fractional derivatives. Linear stability analysis is performed for different values of derivative orders. Results of qualitative analysis are confirmed by numerical simulations of specific partial differential equations. Most attention is paid to two models: a fractional order reaction diffusion system with Bonhoeffer–van der Pol kinetics and to the Brusselator model.

Mathematical modeling of time fractional reaction–diffusion systems

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.