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.

Investigating the Dynamic Behavior of Integer and Noninteger Order System of Predation with Holling’s Response

The paper’s primary objective is to examine the dynamic behavior of an integer and noninteger predator–prey system with a Holling type IV functional response in the Caputo sense. Our focus is on understanding how harvesting influences the stability, equilibria, bifurcations, and limit cycles within this system. We employ qualitative and quantitative analysis methods rooted in bifurcation theory, dynamical theory, and numerical simulation. We also delve into studying the boundedness of solutions and investigating the stability and existence of equilibrium points within the system. Leveraging Sotomayor’s theorem, we establish the presence of both the saddle-node and transcritical bifurcations. The analysis of the Hopf bifurcation is carried out using the normal form theorem. The model under consideration is extended to the fractional reaction–diffusion model which captures non-local and long-range effects more accurately than integer-order derivatives. This makes fractional reaction–diffusion systems suitable for modeling phenomena with anomalous diffusion or memory effects, improving the fidelity of simulations in turn. An adaptable numerical technique for solving this class of differential equations is also suggested. Through simulation results, we observe that one of the Lyapunov exponents has a negative value, indicating the potential for the emergence of a stable-limit cycle via bifurcation as well as chaotic and complex spatiotemporal distributions. We supplement our analytical investigations with numerical simulations to provide a comprehensive understanding of the system’s behavior. It was discovered that both the prey and predator populations will continue to coexist and be permanent, regardless of the choice of fractional parameter.

Passivity of fractional reaction-diffusion systems

This study considers the passivity of fractional reaction-diffusion systems (FRDSs) featuring boundary input-output. The fundamental framework for this exploration involves the application of the Lyapunov-Krasovskii functional method coupled with inequality techniques. The investigation derives sufficient conditions that guarantee both input and output strict passivity in FRDSs. Furthermore, in the presence of fluctuating uncertain parameters within FRDSs, robust passivity is also studied and sufficient conditions are offered. The obtained sufficient condition shows that a larger diffusion coefficient facilitates the achievement of the passivity of FRDSs. Finally, the validity of our theoretical findings is subsequently corroborated through numerical examples.

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.

Nonlocal p-Laplacian involving a nonlinear fractional reaction-diffusion system applied to image restoration

Image restoration has been a longstanding challenge in image processing, involving the enhancement of image content to extract valuable information. Nonlinear models have demonstrated their efficacy in eliminating additive noise, which motivates our goal of investigating a novel nonlocal nonlinear reaction-diffusion model for noise removal. The proposed model integrates a fractional diffusion equation with a nonlinear nonlocal p-Laplace operator, with the fidelity term employing a weak norm to better capture oscillatory patterns and intricate details in textured images. Using the Schauder fixed point theorem, the well-posedness of the proposed model solution is established. The experimental results confirm the effectiveness and efficiency of the proposed model, providing validation for its practical utility.

Boundary stabilisation of fractional reaction-diffusion systems with time-varying delays

This paper examines the boundary stabilisation results for time fractional-order reaction-diffusion systems involving with time-varying delays. The main goal is to design the boundary control for the system by proving the well-posedness of the kernel function using the backstepping method. An invertible Volterra integral transformation is used to convert the considered system into a stable target system. Different from the existing results, the stability results for fractional RDEs are discussed in the sense of the Lyapunov–Krasovskii theory and sufficient conditions are derived with the help of the linear matrix inequality (LMI) approach. Finally, to show the application of the results, the proposed conditions are numerically validated over a time fractional-order reaction-diffusion cellular neural network model.

Mittag–Leffler stabilization for short memory fractional reaction-diffusion systems via intermittent boundary control

Mittag–Leffler stabilization for short memory fractional reaction-diffusion systems via intermittent boundary control

An investigation of global existence of the solution of fractional reaction-diffusion system

An investigation of global existence of the solution of fractional reaction-diffusion system

On Global Existence of the Fractional Reaction-Diffusion System’s Solution

The purpose of this paper is to prove the global existence of solution for one of most significant fractional partial differential system called the fractional reaction-diffusion system. This will be carried out by combining the compact semigroup methods with some L1-estimate methods. Our investigation can be applied to a wide class of fractional partial differential equations even if they contain nonlinear terms in their constructions.

Nonstationary homoclinic solutions for infinite-dimensional fractional reaction-diffusion system with two types of superlinear nonlinearity

This paper is dedicated to studying nonstationary homoclinic solutions with the least energy for a class of fractional reaction-diffusion system \begin{document}$ \begin{eqnarray*} \label{1.1} \left\{\begin{array}{lll} \partial_t u+ (-\Delta)^s u+V(x)u+W(x)v = H_v(t, x, u, v), \\ - \partial_t v + (-\Delta)^s v+V(x)v+W(x)u = H_u(t, x, u, v), \\ |u(t, x)|+|v(t, x)|\rightarrow 0, \ \ \text{as}\ \ |t|+|x|\rightarrow \infty, \end{array} \right. \end{eqnarray*} $\end{document} where $ 0<s<1, \ z = (u, v): \mathbb{R}\times \mathbb{R}^N\rightarrow \mathbb{R}^{2} $, which originate from a wide variety of fields such as theoretical physics, optimal control, chemistry and biology. We obtain ground state solutions of Nehari-Pankov type under mild conditions on the nonlinearity by further developing non-Nehari method with two types of superlinear nonlinearity. If in addition the corresponding functional is even, we also obtain infinitely many geometrically distinct solutions by using some arguments about deformation type and Krasnoselskii genus. Nevertheless, we need to overcome some difficulties: one is that the associated functional is strongly indefinite, the second is due to the absence of strict monotonicity condition, a key ingredient of seeking the ground state solution on suitable manifold, we need some new methods and techniques. The third lies that some delicate analysis are needed for the dropping of classical super-quadratic assumption on the nonlinearity and in verifying the link geometry and showing the boundedness of Cerami sequences.

Finite‐time boundary stabilization of fractional reaction‐diffusion systems

This paper investigates the boundary finite‐time stabilization of fractional reaction‐diffusion systems (FRDSs). First, a distributed controller is designed, and sufficient conditions are obtained to ensure the finite‐time stability (FTS) of FRDSs under the designed controller. Then, a boundary controller is presented to achieve the FTS. By virtue of Lyapunov functional method and inequality techniques, sufficient conditions are presented to ensure the FTS of FRDSs via the designed boundary controller. The effect of diffusion term of FRDSs on the FTS is also investigated. Both Neumann and mixed boundary conditions are considered. Moreover, the robust finite‐time stabilization of uncertain FRDSs is studied when there are uncertainties in the system's coefficients. Under the designed boundary controller, sufficient conditions are presented to guarantee the robust FTS of uncertain FRDSs. Finally, numerical examples are presented to verify the effectiveness of our theoretical results.

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.

Homotopy Perturbation Method for Solving Nonlinear Fractional Reaction Diffusion Systems

This study presents the Homotopy Perturbation Method (HPM) for nonlinear fractional reaction diffusion systems, the fractional derivatives are described in the caputo's' fractional operator. The study focus on three systems of fractional reaction diffusion equations in one, two and three dimensions, in this method, the solution considered as the sum of an infinite series. Which converges rapidly to exact solution. The Homotopy Perturbation Method is no need to use Adomian's polynomials to calculate the nonlinear terms; we test the proposed method to solve nonlinear fractional systems of redaction diffusion equations in one dimension, two dimensions and three dimensions. To show the efficiency and accuracy of this method, we compared the results of the fractional derivatives orders with ordinary derivative order index <i>α</i>₁=<i>α</i>₂=1 for nonlinear fractional reaction diffusion systems. Approximate solutions for different values of fractional derivatives index <i>α</i>₁=0.5 and <i>α</i>₂=0.5 together with non-fractional derivative index α₁=1 and <i>α</i>₂=1 and absolute errors are represented graphically in two and three dimensions. In addition, the graphical represented the solutions, which had been given by MATLAB program. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional systems of partial differential equations over existing methods.

Nonstationary homoclinic orbit for an infinite-dimensional fractional reaction-diffusion system

&lt;p style='text-indent:20px;'&gt;This paper study nonstationary homoclinic-type solutions for a fractional reaction-diffusion system with asymptotically linear and local super linear nonlinearity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite, the second lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is not super quadratic at infinity globally. These enable us to develop a direct approach and new tricks to overcome the difficulties. We establish the existence of homoclinic orbit under some weak assumptions on nonlinearity.&lt;/p&gt;

Analysis and pattern formation scenarios in the superdiffusive system of predation described with Caputo operator

The concept of a fractional derivative is introduced to the predator-prey system to describe the species anomalous superdiffusive process. To achieve this, a new class of predator-prey model with the Beddington-DeAngelis functional response is formulated in the sense of the Caputo fractional order operator. This work aims to give a mathematical basis for computational studies of a two-variable fractional reaction-diffusion system in one and two dimensions from biological and numerical perspectives. As a result, some details of the local and global dynamics of the reaction-diffusion system are provided by using the idea of the linear stability analysis and well-known dynamical systems theory to derive conditions on the parameters which can guarantee biologically meaningful equilibria also serve as a guide in ensuring the correct choice of parameters when numerically experimenting with the solutions of the full fractional reaction-diffusion model. The behavior of the new dynamical system is examined for both diffusive and non-diffusive systems at different instances of fractional order.

Boundary output feedback stabilization for spacial multi‐dimensional coupled fractional reaction–diffusion systems

AbstractThis paper studies the boundary feedback stabilization for spacial multi‐dimensional coupled fractional reaction–diffusion systems with non‐collocated or collocated outputs in the case that the system state is unmeasurable. By employing the backstepping method, we introduce a target system and analyze its Mittag–Leffler stability. For each pair of sides of the region boundary, assuming that system state is hinged on one side, while the controller is designed on the opposite side. These boundary controllers work together to achieve the state feedback Mittag–Leffler stabilization of the considered system. In addition, for both kinds of outputs, feedback controllers are also established to achieve the asymptotical stability of the corresponding closed‐loop system. Two numerical experiments are carried out to illustrate our results.

Stable finite difference method for fractional reaction\u2013diffusion equations by compact implicit integration factor methods

In this paper we propose a stable finite difference method to solve the fractional reaction–diffusion systems in a two-dimensional domain. The space discretization is implemented by the weighted shifted Grünwald difference (WSGD) which results in a stiff system of nonlinear ordinary differential equations (ODEs). This system of ordinary differential equations is solved by an efficient compact implicit integration factor (cIIF) method. The stability of the second order cIIF scheme is proved in the discrete L^{2}-norm. We also prove the second-order convergence of the proposed scheme. Numerical examples are given to demonstrate the accuracy, efficiency, and robustness of the method.

Analysis and numerical simulation of cross reaction–diffusion systems with the Caputo–Fabrizio and Riesz operators

AbstractThe evolutionary dynamics of cross‐reaction–diffusion equations of predator–prey type are investigated in the sense of fractional operator. In the models, we replace the classical time and spatial derivatives with the Caputo–Fabrizio and Riesz fractional derivatives, respectively. The nature of the resulting problem (is nonlinear, nonlocal, and nonsingular) do not either admit a closed form solution, while in most cases the analytical solution is too involved to be useful. As a result, there is need to provide a reliable numerical scheme that can approximate these derivatives in time and space. Hence, we formulate an approximation scheme with second‐order convergence rate for the time‐Caputo–Fabrizio fractional operator of order 0 &lt; α ≤ 1 and L1 formula for the Riesz fractional derivative of order 1 &lt; β ≤ 2 in space. As a case study, we consider two examples of strongly coupled cross fractional reaction–diffusion systems describing the interaction between two individual species that prey on the other one. We examine the system for stability analysis and establish the condition for the occurrence of Turing instability. The complexity of the dynamics of time–space cross fractional reaction–diffusion systems is theoretically studied and numerically in one and two dimensions for some instances of fractional orders.

Global solutions for a strongly coupled fractional reaction-diffusion system in Marcinkiewicz spaces

Abstract We prove the existence of solutions to the Cauchy problem for a strongly coupled semilinear reaction-diffusion system in Marcinkiewicz spaces L ( p 1 , ∞ ) × L ( p 2 , ∞ ) . The exponents p 1 , p 2 are chosen in a way that allows us to prove the existence of self-similar for this system. We present a fractional version of Yamazaki’s inequality, an essential tool that potentially applies to other fractional-in-time PDEs.