Reaction-diffusion Research Articles

Stability analysis and numerical simulations of the infection spread of epidemics as a reaction–diffusion model

This paper presents a spatiotemporal reaction–diffusion model for epidemics to predict how the infection spreads in a given space. The model is based on a system of partial differential equations with the Neumann boundary conditions. First, we study the existence and uniqueness of the solution of the model using the semigroup theory and demonstrate the boundedness of solutions. Further, the proposed model's basic reproduction number is calculated using the eigenvalue problem. Moreover, the dynamic behavior of the disease‐free steady states of the model for is investigated. The uniform persistence of the model is also discussed. In addition, the global asymptotic stability of the endemic steady state is examined. Finally, the numerical simulations validate the theoretical results.

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.

Distributed sliding mode control for a class of impulsive uncertain delayed partial differential equations via sliding mode compensator approach

The distributed sliding mode control (SMC) method of delayed partial differential equations (PDEs)with information of periodical impulse and reaction–diffusion terms (RDTs) is discussed in the article. Firstly, by adopting the sliding mode compensator approach, the switching function including impulsive effects is constructed, which is continuous along the system states. The sliding motion equation is then derived, and its parameters depend on that of sliding mode compensator.Via the direct linear matrix inequality method(DLIM), the exponential stability of sliding motion is analyzed. The proposed distributed SMC strategy can eliminate the impulsive effects and drive the closed-loop state onto a specific sliding manifold in limited time. Finally, a numerical simulation is presented to certify the availability of the proposed scheme.

Stability Analysis and Hopf Bifurcation for the Brusselator Reaction–Diffusion System with Gene Expression Time Delay

This paper investigates the effect of a gene expression time delay on the Brusselator model with reaction and diffusion terms in one dimension. We obtain ODE systems analytically by using the Galerkin method. We determine a condition that assists in showing the existence of theoretical results. Full maps of the Hopf bifurcation regions of the stability analysis are studied numerically and theoretically. The influences of two different sources of diffusion coefficients and gene expression time delay parameters on the bifurcation diagram are examined and plotted. In addition, the effect of delay and diffusion values on all other free parameters in this system is shown. They can significantly affect the stability regions for both control parameter concentrations through the reaction process. As a result, as the gene expression time delay increases, both control concentration values increase, while the Hopf points for both diffusion coefficient parameters decrease. These values can impact solutions in the bifurcation regions, causing the region of instability to grow. In addition, the Hopf bifurcation points for the diffusive and non-diffusive cases as well as delay and non-delay cases are studied for both control parameter concentrations. Finally, various examples and bifurcation diagrams, periodic oscillations, and 2D phase planes are provided. There is close agreement between the theoretical and numerical solutions in all cases.

Nonlinear Stability and Asymptotic Behavior of Periodic Wave Trains in Reaction–Diffusion Systems Against Cub\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\ extrm{ub}}$$\\end{document}-perturbations

We present a nonlinear stability theory for periodic wave trains in reaction–diffusion systems, which relies on pure L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document}-estimates only. Our analysis shows that localization or periodicity requirements on perturbations, as present in the current literature, can be completely lifted. Inspired by previous works considering localized perturbations, we decompose the semigroup generated by the linearization about the wave train and introduce a spatio-temporal phase modulation to capture the most critical dynamics, which is governed by a viscous Burgers’ equation. We then aim to close a nonlinear stability argument by iterative estimates on the corresponding Duhamel formulation, where, hampered by the lack of localization, we must rely on diffusive smoothing to render decay of the semigroup. However, this decay is not strong enough to control all terms in the Duhamel formulation. We address this difficulty by applying the Cole–Hopf transform to eliminate the critical Burgers’-type nonlinearities. Ultimately, we establish nonlinear stability of diffusively spectrally stable wave trains against Cub\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\ extrm{ub}}$$\\end{document}-perturbations. Moreover, we show that the perturbed solution converges to a modulated wave train, whose phase and wavenumber are approximated by solutions to the associated viscous Hamilton–Jacobi and Burgers’ equation, respectively.

Insights into reaction–diffusion behaviors of acetylene selective hydrogenation on pellet catalysts

AbstractIn this work, a particle‐resolved computational fluid dynamics model of the acetylene hydrogenation process is developed to investigate the effects of catalyst particle structures on the reaction–diffusion behaviors aiming to improve selectivity toward target ethylene. The effects of packing structures on mass and heat transfer are explored by employing particles with varying shapes, wall thicknesses, and external diameters. The simulation results reveal that decreasing diffusion paths and elevating reactor bed temperature will enhance ethylene selectivity, and cylinder and Raschig ring packing structures exhibit the lowest and highest ethylene selectivity of 35.6% and 48.9% at 70% of acetylene conversion, respectively. Reducing wall thicknesses of Raschig ring particles facilitates the diffusion of generated ethylene from the interior zone of catalysts but concurrently inhibits the conversion of acetylene to ethylene. The Raschig ring catalyst particle with 1.9 mm of wall thickness and 3.5 mm of external diameter is finally revealed to exhibit the highest ethylene selectivity.

Novel Numerical Investigation of Reaction Diffusion Equation Arising in Oil Price Modeling

Consideration is given to a reaction–diffusion free boundary value problem with one or two turning points arising in oil price modeling. First, an exact (analytical) solution to the reduced problem (i.e., no diffusion term) was obtained for some given parameters. The space–time Chebyshev pseudospectral and superconsistent Chebyshev collocation method is proposed for both reaction diffusion (RDFBP) and reduced free boundary value problem. Error bounds on the discrete L2–norm and Sobolev norm (Hp) are presented. Adaptively graded intervals were introduced and used according to the value of turning points to avoid the twin boundary layers phenomena. Excellent convergent (spectrally) and stable results for some special turning points were obtained for both reduced and RDFBP equations on an adaptively graded interval and this has been documented for the first time.

Almost automorphic behaviours of nonlocal stochastic fuzzy Cohen–Grossberg lattice neural networks

Discrete-time Cohen–Grossberg neural networks (CGNNs) play important role in feedback neural networks. The existing literatures of CGNNs only regarded integral order discrete-time models without other variables. This paper builds the lattice model for nonlocal stochastic fuzzy CGNNs with reaction diffusions by employing the finite difference and Mittag–Leffler time Euler difference techniques. Further, the existence of a unique bounded almost automorphic sequence solution in distribution and global exponential convergence in the mean-square sense to the achieved difference model are investigated. At last, by using the tool of Matlab and time Euler difference for the Brownian motions, an illustrative example with simulations is used to show the feasible of the works of the current paper. We think that this work can contribute to achieve some effective real tasks, e.g. content-addressable memory, auto-association, dynamic reconstruction of a chaotic process, etc.

Spreading speeds in an asymptotic autonomous system with application to an epidemic model

This article studies the initial value problem in an asymptotic autonomous reaction–diffusion system. Namely, when the time goes to infinity, these parameters converge to constants. The system may be nonmonotonic. With different decaying initial conditions, the leftward and rightward spreading speeds are estimated by constructing auxiliary functions and systems. As an application, we present the propagation properties in a diffusive epidemic model with different decaying initial values, which does not satisfy the classical comparison principle of mixed quasimonotone systems.

A hybrid kernel-based meshless method for numerical approximation of multidimensional Fisher’s equation

We propose and analyze a meshless method of lines by considering some hybrid radial kernels. These hybrid kernels are constructed by linearly combining infinite smooth radial functions to piecewise smooth radial functions; which are then used for spatial approximation on trial spaces spanned by translates of positive definite radial functions. After spatial approximation, a high-order ODE solver is invoked for efficient and stable time-integration of the resultant semi-discrete system of ordinary differential equations (ODEs). Unlike the mesh-based method of lines, the proposed method works for arbitrary scattered data points and is equally effective for problems over non-rectangular domains. The proposed method is tested on one-, two- and three-dimensional reaction–diffusion Fisher equation for its numerical stability, accuracy, and efficiency against the contemporary meshless and mesh-based methods. The economical computational cost, improved accuracy, eigenvalues stability, and well-conditioning of system matrices are observed against RBF collocation and RBF-PS methods.

Hybrid control of Turing instability and bifurcation for spatial-temporal propagation of computer virus

ABSTRACT In this era of information technology, information leakage and file corruption due to computer virus intrusion have been serious issues. How to detect and prevent the spread of the computer virus is the major challenge we are facing now. To target this problem, a class of virus propagation models with hybrid control scheme are formulated to investigate the dynamic evolution and prevention from a spatial-temporal perspective in this paper. Diffusion-induced Turing instability is detected in response to the computer virus propagation. The introduction of hybrid control scheme can effective suppress Turing instability and turn the propagation system back to a stable state. And then, the time delay is selected as the bifurcation parameter. If the time delay exceeds the bifurcation threshold, the propagation will be destabilised and a Hopf bifurcation will occur. The hybrid control tactic can not only regulate the occurrence of Hopf bifurcation well, but also optimise the properties of bifurcating period solutions. In the end, the correctness and validity of the theoretical results are verified via numerical simulations.

On stability of a reaction diffusion system described by difference equations

ABSTRACT This work investigates the dynamics of discrete reaction-diffusion Gierer-Meinhardt system as mathematical models of biological pattern formation. We study the system's local asymptotic behaviour with and without the diffusion once developing the discrete integer variant of the well-known Gierer-Meinhardt model and proving that the model has a unique equilibrium. The requirements for the steady-state solution's local and global stability are found with the help of relevant approaches and the Lyapunov technique. Two large biological models and simulations are used throughout the work to validate the utility of the suggested technique.

On the generalized time fractional reaction–diffusion equation: Lie symmetries, exact solutions and conservation laws

In this paper, Lie symmetry analysis method is applied to the generalized time fractional reaction–diffusion equation. We obtain a conditional symmetric group and some conservation laws of the governing equation. The obtained Lie symmetries are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative. Furthermore, we obtained asymptotic stable solutions and convergent power series solutions for the reduced equations. The dynamic behavior of these exact solutions is presented graphically.

Formation Mechanism of Ti–Si Multi-Layer Coatings on the Surface of Ti–6Al–4V Alloy

Titanium alloys are widely used in aerospace applications due to their high specific strength and exceptional corrosion resistance. In this study, a silicide coating with a multi-layer structure was designed and prepared via a pack cementation process to improve the high-temperature oxidation resistance of titanium alloy. A new theory based on the Le Chatelier’s principle is proposed to explain the generation mechanism of active Si atoms. Taking the chemical potential as a bridge, a functional model of the relationship between the diffusion driving force and the change in the Gibbs free energy of reaction diffusion is established. Experimental results indicate that the depth of the silicide coating increases with the siliconization temperature (1000–1100 °C) and time (0–5 h). The multi-layer coating prepared at 1075 °C for 3 h exhibits a thick and dense structure with a thickness of 23.52 μm. This coating consists of an outer layer of TiSi2 (9.40 μm), a middle layer of TiSi (3.36 μm), and an inner layer of Ti5Si3 (10.76 μm). Under this preparation parameter, increasing the temperature or prolonging the holding time will cause the outward diffusion flux of atoms in the substrate to be much larger than the diffusion flux of silicon atoms to the substrate, thus forming pores in the coating. The calculated value of the diffusion driving force FTiSi = 2.012S is significantly smaller than that of FTiSi2 = 13.120S and FTi5Si3 = 14.552S, which perfectly reveals the relationship between the thickness of each layer in the Ti–Si multi-layer coating.

An SIS reaction–diffusion model with spatial/behavioral heterogeneity

An SIS reaction–diffusion model with spatial/behavioral heterogeneity

Modeling of the dissolution phenomena of solid particles into an infinitely large medium from the analytical solutions of reaction–diffusion equations

Mathematical solutions of the concentration of dissolved component in surrounding fluid were obtained for analyses of the dissolution phenomena of spherical, cylindrical, and slab-type particles. Calculation of the surface flux enabled material balance equations to be derived for the prediction of transient particle size. For the spherical particle, the resulting differential equation was solved numerically to compare with the analytical solution. For the cylindrical particle, the Weber transform was applied to predict the radius of the infinitely long cylinder as a function of time. After obtaining an analytical solution for the thickness of the dissolving slab, the result was compared with those of the sphere and cylinder. The spherical particle exhibited the shortest complete dissolution time (CDT), due to having the greatest surface-to-volume ratio. When external film resistance was important for the mass transfer, the Biot number was included in the governing equation, and to predict the particle size during dissolution, the concentration in the fluid phase could be derived by the combination of variables, or the Weber-type transform. When first-order or zero-order reaction occurs in the surrounding fluid, the concentration could also be predicted by the Laplace transform, and the particle size could be calculated for the solid particles. Similar to the simple diffusion process, CDT was predicted to increase in the order sphere < cylinder < slab. The experimental data of the dissolution of polymeric fiber was compared with the modeling result to estimate the diffusivity and solubility factor, which varied with the pH of the surrounding fluid.

Turing Instability and Spatial Pattern Formation in a Model of Urban Crime

A nonlinear crime model is generalized by introducing self- and cross-diffusion terms. The effect of diffusion on the stability of non-negative constant steady states is applied. In particular, the cross-diffusion-driven instability, called Turing instability, is analyzed by linear stability analysis, and several Turing patterns driven by the cross-diffusion are studied through numerical investigations. When the Turing–Hopf conditions are satisfied, the type of instability highlighted in the ODE model persists in the PDE system, still showing an oscillatory behavior.

Entire solutions with and without radial symmetry in balanced bistable reaction–diffusion equations

AbstractLet $$n\ge 2$$ n ≥ 2 be a given integer. In this paper, we assert that an n-dimensional traveling front converges to an $$(n-1)$$ ( n - 1 ) -dimensional entire solution as the speed goes to infinity in a balanced bistable reaction–diffusion equation. As the speed of an n-dimensional axially symmetric or asymmetric traveling front goes to infinity, it converges to an $$(n-1)$$ ( n - 1 ) -dimensional radially symmetric or asymmetric entire solution in a balanced bistable reaction–diffusion equation, respectively. We conjecture that the radially asymmetric entire solutions obtained in this paper are associated with the ancient solutions called the Angenent ovals in the mean curvature flows.

Designing Urchin-Like S/SiO2 with Regulated Pores Toward Ultra-Fast Room Temperature Sodium-Sulfur Battery.

Captured by high theoretical capacity and low-cost, Sodium-Sulfur (Na-S) batteries have been deemed as promising energy-storage systems. However, their electrochemical properties, containing both cycling and rate properties, still suffer from the notorious "shuttle effect" of polysulfide. Herein, through the effective regulation of pore sizes, a series of S/SiO2 cathode materials are obtained. Benefitting from the abundant pore channels of SiO2 particles, the sulfur loading is as high as 76.3%. Importantly, a suitable pore size can lead to adequate reaction and rapid diffusion behaviors, resulting in excellent electrochemical performances. Specifically, at 2.0Ag-1, the initial capacity of the as-optimized sample can be up to 1370.6mAhg-1. Surprisingly, even after 1050 cycles, it could achieve a high reversible capacity of 1280.8mAhg-1 with an attenuation rate of 0.089%. At 5.0Ag-1, after 500 cycles, the capacity can still remain ≈ 1132.6mAhg-1 (capacity retention rate, 97.5%). Given this, the work is anticipated to offer an effective strategy for advanced electrodes for Na-S batteries.

Event-triggered passivity and synchronization of multiple derivative coupled reaction–diffusion neural networks

The paper studies the event-triggered passivity and synchronization problems for multiple derivative coupled reaction–diffusion neural networks. Two types of multiple derivative coupled reaction–diffusion neural networks, with and without time-varying delay, are considered. According to the characteristics of multiple derivative coupled reaction–diffusion neural networks, an event-triggered condition is constructed. By imposing appropriate Lyapunov functionals and inequality techniques, several passivity conditions are established for these two networks under the designed event-triggered controllers. Then, synchronization criteria of these two networks are derived based on the relationship between output-strict passivity and synchronization. Moreover, the non-occurrence of the Zeno behavior in the proposed control is proved. Numerical example is illustrated to verify the effectiveness of the proposed criteria. Finally, theoretical results and the associated image encryption application are validated through numerical simulations and statistical analyses.