The Discrete SIR Epidemic Reaction–Diffusion Model: Finite‐Time Stability and Numerical Simulations

This paper is centered on the problem of delay-dependent finite-time stability and stabilization for a class of continuous system with additive time-varying delays. Firstly, based on a new Lyapunov---Krasovskii-like function (LKLF), which splits the whole delay interval into some proper subintervals, a set of delay-dependent finite-time stability conditions, guaranteeing that the state of the system does not exceed a given threshold in fixed time interval, are derived in form of linear matrix inequalities. In particular, to obtain a less conservative result, we take the LKLF as a whole to examine its positive definite which can slack the requirements for Lyapunov matrices and reduce the loss information when estimating the bound of the function. Further, based on the results of finite-time stability, sufficient conditions for the existence of a state feedback finite-time controller, guaranteeing finite-time stability of the closed-loop system, are obtained and can be solved by using some standard numerical packages. Finally, some numerical examples are provided to demonstrate the less conservative and the effectiveness of the proposed design approach.

The aim of this work is to design two novel implicit and explicit finite difference (FD) schemes to solve SIR (susceptible, infected and recovered) epidemic reaction–diffusion system with modified saturated incidence rate. Since this model is based on population dynamics, therefore solution of the continuous system possesses the positivity property. The proposed finite difference schemes retain the positivity property of sub population which is an essential feature in population dynamics. Von Neumann stability analysis reveals that proposed FD schemes are unconditionally stable. It is verified with the help of Taylor's series expansion that proposed FD schemes are consistent. The proposed implicit scheme is unconditionally consistent, i.e. for . On the other hand the proposed explicit scheme gives conditional consistency for . The proposed FD schemes are compared with two other FD schemes, i.e. forward Euler and Crank Nicolson scheme. Simulations are performed for the verification of all the attributes for the underlying FD schemes. Furthermore, stability of the reaction diffusion system is discussed by applying Routh–Hurwitz criteria. Bifurcation values of infection coefficient are also obtained from Routh–Hurwitz condition.

This study discusses the performance comparison of three numerical approaches, namely the Shooting method, the Finite Difference Method (FDM), and the Collocation method, in solving Boundary Value Problems (BVP) for three categories of problems: linear non-stiff, linear stiff, and non-linear. Each type of problem is tested with two types of boundary conditions, namely Dirichlet and Neumann. The evaluation is based on two main criteria: accuracy, measured using error norms with respect to the exact solution, and computational efficiency, quantified in terms of CPU execution time. The results show that in the non-stiff case with Dirichlet boundary conditions, the Shooting methods based on LSODA and RK5 provide very high accuracy with good efficiency, while the Finite Difference Method excels in efficiency but is slightly inferior in accuracy. Under Neumann boundary conditions, the Finite Difference Method tends to be less accurate, whereas the Collocation method delivers very good accuracy but with relatively lower efficiency. For stiff problems, the Shooting method maintains high accuracy, while the Finite Difference and Collocation methods show varying performance depending on the type of boundary condition. In the non-linear case, the Shooting method becomes the most accurate option, although with slightly lower efficiency compared to the Finite Difference Method. These findings provide practical guidance in selecting appropriate numerical methods for BVPs based on problem characteristics and boundary conditions.

This study assesses the one-dimensional (1D) consolidation of unsaturated soil subjected to time-dependent loading based on the 1D consolidation theories of unsaturated soil. The differential quadrature method (DQM) is used to produce a general solution that considers various boundary conditions, various initial pore-water and pore-air distributions, and complex time-dependent loading. A special case in which the analytical solution is available in the literature is used for verification and accuracy analysis. It is found that the DQM solution can deliver more accurate results compared with the finite-difference method with a small number of sampling points. The general solution can avoid cumbersome computations in solving eigenequations encountered with the analytical solution. In addition, the proposed solution is more suitable for practical engineering because of its generality in complex initial, boundary, and loading conditions. Finally, the characteristics of the 1D consolidation of unsaturated soils under various initial pore pressure distributions, boundary conditions, and complex time-dependent loading conditions are investigated, and it is found that the initial and boundary conditions have a significant influence on the consolidation of unsaturated soils.

A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains

Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Sufficient conditions for finite-time stability have recently been developed in the literature for discrete-time dynamical systems. In this article, we build on these results to develop a framework for addressing the problem of optimal nonlinear analysis and feedback control for finite-time stability and finite-time stabilization for nonlinear discrete-time controlled dynamical systems. Finite-time stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that satisfies a difference inequality involving fractional powers and a minimum operator. This Lyapunov function can clearly be seen to be the solution to a difference equation that corresponds to a steady-state form of the Bellman equation, and hence, guaranteeing both finite-time stability and optimality. Finally, a numerical example is presented to demonstrate the efficacy of the proposed finite-time discrete stabilization framework.

In this paper, finite-time stability and stabilization problems for switched linear systems are discussed. Firstly, the concept of finite-time stability is extended to switched linear systems. Then, based on the state transition matrix of the system, a necessary and sufficient condition for finite-time stability of switched linear systems is presented. For ease of reference, some sufficient conditions under which the switched linear systems are finite-time stable and uniformly finite-time stable are given by virtue of matrix inequalities. Moreover, stabilizing state feedback controllers and a class of switching signals with average dwell-time are designed in detail to solve finite-time stabilization problem. The main results are proved by using the multiple Lyapunov-like functions and common Lyapunov-like function respectively. Finally, two examples are employed to verify the efficiency of the proposed method.

A new type of thin plate model and the related nonclassical boundary value problems were established within the framework of strain gradient and velocity gradient elasticity. The closed-form solutions of deflections and free vibrational frequencies of a simply supported plate resting on an elastic foundation were obtained. The results of the present model agree well with those predicted by the molecular dynamics. Numerical results show that, the elastic foundation and the strain gradient parameter have a stiffness-hardening effect, while the velocity gradient parameter has a stiffness-softening effect. The proposed boundary value problems are of great significance to the study of the mechanical behaviors of plates under complex boundary conditions and external loadings. Furthermore, it will be useful for developing effective numerical methods such as the finite element method, the finite difference method and the Garlerkin method.

Simulating surface conditions by solving the wave equation of a sucker-rod string is the theoretical basis of a sucker-rod pumping system. To overcome the shortcomings of the conventional finite difference method and analytical solution, this work describes a novel hybrid method that combines the analytical solution with the finite difference method. In this method, an analytical solution of the tapered rod wave equation with a recursive matrix form based on the Fourier series is proposed, a unified pumping condition model is established, a modified finite difference method is given, a hybrid strategy is established, and a convergence calculation method is proposed. Based on two different types of oil wells, the analytical solutions are verified by comparing different methods. The hybrid method is verified by using the finite difference method simulated data and measured oil data. The pumping speed sensitivity and convergence of the hybrid method are studied. The results show that the proposed analytical solution has high accuracy, with a maximum relative error relative to that of the classical finite difference method of 0.062%. The proposed hybrid method has a high simulation accuracy, with a maximum relative area error relative to that of the finite difference method of 0.09% and a maximum relative area error relative to measured data of 1.89%. Even at higher pumping speeds, the hybrid method still has accuracy. The hybrid method in this paper is convergent. The introduction of the finite difference method allows the hybrid method to more easily converge. The novelty of this work is that it combines the advantages of the finite difference method and the analytical solution, and it provides a convergence calculation method to provide guidance for its application. The hybrid method presented in this paper provides an alternative scheme for predicting the behavior of sucker-rod pumping systems and a new approach for solving wave equations with complex boundary conditions.

The solution of solute transport problem in an aquifer with suitable boundary conditions has been dealt by various analytical methods in the past. But, the analytical approach becomes more difficult to apply when either dealing with complex boundary conditions or higher order solute transport problems. The difficulty may be reduced by handling the problem by numerical approaches as in the present paper, forward in time centered in space (FTCS) finite-difference scheme is used to solve the three-dimensional advection-dispersion equation (ADE) with Dirichlet and Neumann boundary conditions. The Dirichlet boundary conditions are taken temporally dependent. The numerical solution is obtained graphically with the help of MATLAB software package, and further under a special case, the numerical solution obtained by FTCS scheme is validated with the solution obtained by PDEtool which is based on the finite-element method.

To model bio-chemical reaction systems with diffusion one can either use stochastic, microscopic reaction–diffusion master equations or deterministic, macroscopic reaction–diffusion system. The connection between these two models is not only theoretically important but also plays an essential role in applications. This paper considers the macroscopic limits of the chemical reaction–diffusion master equation for first-order chemical reaction systems in highly heterogeneous environments. More precisely, the diffusion coefficients as well as the reaction rates are spatially inhomogeneous and the reaction rates may also be discontinuous. By carefully discretizing these heterogeneities within a reaction–diffusion master equation model, we show that in the limit we recover the macroscopic reaction–diffusion system with inhomogeneous diffusion and reaction rates.

ABSTRACTTwo-player zero-sum differential games are addressed within the framework of state-feedback finite-time partial-state stabilisation of nonlinear dynamical systems. Specifically, finite-time partial-state stability of the closed-loop system is guaranteed by means of a Lyapunov function, which we prove to be the value of the game. This Lyapunov function verifies a partial differential equation that corresponds to a steady-state form of the Hamilton–Jacobi–Isaacs equation, and hence guarantees both finite-time stability with respect to part of the system state and the existence of a saddle point for the system's performance measure. Connections to optimal regulation for nonlinear dynamical systems with nonlinear-nonquadratic cost functionals in the presence of exogenous disturbances and parameter uncertainties are also provided. Furthermore, we develop feedback controllers for affine nonlinear systems extending an inverse optimality framework tailored to the finite-time partial-state stabilisation problem. Finally, two illustrative numerical examples show the applicability of the results proven.

This paper delves into a comprehensive analysis of a generalized impulsive discrete reaction–diffusion system under periodic boundary conditions. It investigates the behavior of reactant concentrations through a model governed by partial differential equations (PDEs) incorporating both diffusion mechanisms and nonlinear interactions. By employing finite difference methods for discretization, this study retains the core dynamics of the continuous model, extending into a discrete framework with impulse moments and time delays. This approach facilitates the exploration of finite-time stability (FTS) and dynamic convergence of the error system, offering robust insights into the conditions necessary for achieving equilibrium states. Numerical simulations are presented, focusing on the Lengyel–Epstein (LE) and Degn–Harrison (DH) models, which, respectively, represent the chlorite–iodide–malonic acid (CIMA) reaction and bacterial respiration in Klebsiella. Stability analysis is conducted using Matlab’s LMI toolbox, confirming FTS at equilibrium under specific conditions. The simulations showcase the capacity of the discrete model to emulate continuous dynamics, providing a validated computational approach to studying reaction-diffusion systems in chemical and biological contexts. This research underscores the utility of impulsive discrete reaction-diffusion models for capturing complex diffusion–reaction interactions and advancing applications in reaction kinetics and biological systems.

Finite difference reaction–diffusion systems with coupled boundary conditions and time delays

Mesoscopic methods serve as a pivotal link between the macroscopic and microscopic scales, offering a potent solution to the challenge of balancing physical accuracy with computational efficiency. Over the past decade, significant progress has been made in the application of the discrete Boltzmann method (DBM), which is a mesoscopic method based on a fundamental equation of nonequilibrium statistical physics (i.e., the Boltzmann equation), in the field of nonequilibrium fluid systems. The DBM has gradually become an important tool for describing and predicting the behavior of complex fluid systems. The governing equations comprise a set of straightforward and unified discrete Boltzmann equations, and the choice of their discrete format significantly influences the computational accuracy and stability of numerical simulations. In a bid to bolster the reliability of these simulations, this paper utilizes the finite volume method as a solution for handling the discrete Boltzmann equations. The finite volume method stands out as a widely employed numerical computation technique, known for its robust conservation properties and high level of accuracy. It excels notably in tackling numerical computations associated with high-speed compressible fluids. For the finite volume method, the value of each control volume corresponds to a specific physical quantity, which makes the physical connotation clear and the derivation process intuitive. Moreover, through the adoption of suitable numerical formats, the finite volume method can effectively minimize numerical oscillations and exhibit strong numerical stability, thus ensuring the reliability of computational results. Particularly, the MUSCL format where a flux limiter is introduced to improve the numerical robustness is adopted for the reconstruction in this paper. Ultimately, the DBM utilizing the finite volume method is rigorously validated to assess its proficiency in addressing flow issues characterized by pronounced discontinuities. The numerical experiments encompass scenarios involving shock waves, Lax shock tubes, and acoustic waves. The results demonstrate the method's precise depiction of shock wave evolution, rarefaction waves, acoustic phenomena, and material interfaces. Furthermore, it ensures the conservation of mass, momentum, and energy within the system, as well as accurately measures the hydrodynamic and thermodynamic nonequilibrium effects of the fluid system. Compared with the finite difference method, the finite volume method is also more convenient and flexible in dealing with boundary conditions of different geometries, and can be adapted to a variety of systems with complex boundary conditions. Consequently, the finite volume method further broadens the scope of DBM in practical applications.