Non-standard finite difference method applied to an initial boundary value problem describing hepatitis B virus infection

In this paper, two non-standard finite difference (NSFD) schemes are proposed for a mathematical model of hepatitis B virus (HBV) infection with spatial dependence. The dynamic properties of the obtained discretized systems are completely analyzed. Relying on the theory of M-matrix, we prove that the proposed NSFD schemes is unconditionally positive. Furthermore, we establish that the NSFD method used preserves all constant steady states of the corresponding continuous initial boundary value problem (IBVP) model. We prove that the conditions for those equilibria to be asymptotically stable are consistent with the continuous IBVP model independently of the numerical grid size. The global asymptotical properties of the HBV-free equilibrium of the proposed NSFD schemes are derived via the construction of a suitable discrete Lyapunov function, and coincides with the continuous system. This confirms that the discretized models are dynamically consistent since they maintain essential properties of the corresponding continuous IBVP model. Finally, numerical simulations are performed from which it is demonstrated that the proposed NSFD method is advantageous over the standard finite difference (SFD) method.     

A non-standard finite difference scheme for a diffusive HBV infection model with capsids and time delay

In this paper, a non-standard finite difference (NSFD) scheme for a delayed diffusive hepatitis B virus (HBV) infection model with intracellular HBV DNA-containing capsids is proposed. Dynamic consistency of this NSFD scheme is achieved by showing that the scheme preserves the non-negativity and boundedness of the solutions and the global stability of the homogeneous steady states of the corresponding continuous model without any restriction on spatial and temporal grid sizes. We prove the global stability of the steady states by constructing suitable discrete Lyapunov functions.     

Dynamically consistent discrete metapopulation model

In this paper we construct nonstandard difference schemes, which are dynamically consistent with a metapopulation model formulated by Keymer et al. in 2000, i.e. preserve all dynamical properties of the differential equations of the model. These properties are: monotone convergence, boundedness, local asymptotic stability and especially, global stability of equilibria and non-periodicity of solutions. Numerical examples confirm the obtained theoretical results of the properties of the constructed difference schemes.     

On a new fractional-order Logistic model with feedback control

In this paper, we formulate and analyze a new fractional-order Logistic model with feedback control, which is different from a recognized mathematical model proposed in our very recent work. Asymptotic stability of the proposed model and its numerical solutions are studied rigorously. By using the Lyapunov direct method for fractional dynamical systems and a suitable Lyapunov function, we show that a unique positive equilibrium point of the new model is asymptotically stable. As an important consequence of this, we obtain a new mathematical model in which the feedback control variables only change the position of the unique positive equilibrium point of the original model but retain its asymptotic stability. Furthermore, we construct unconditionally positive nonstandard finite difference(NSFD) schemes for the proposed model using the Mickens' methodology. It is worth noting that the constructed NSFD schemes not only preserve the positivity but also provide reliable numerical solutions that correctly reflect the dynamics of the new fractional-order model. Finally, we report some numerical examples to support and illustrate the theoretical results. The results indicate that there is a good agreement between the theoretical results and numerical ones.     

Nonstandard finite difference schemes for a class of generalized convection–diffusion–reaction equations

In this work, a class of nonstandard finite difference (NSFD) schemes are proposed to approximate the solutions of a class of generalized convection–diffusion–reaction equations. First, in the case of no diffusion, two exact finite difference schemes are presented using the method of characteristics. Based on these two exact schemes, a class of exact schemes are presented by introducing a parameter α. Second, since the forms of these exact schemes are so complicated that they are not convenient to use, a class of NSFD schemes are derived from the exact schemes using numerical approximations. It follows that, under certain conditions about denominator function of time‐step sizes, these NSFD schemes are elementary stable and the solutions are positive and bounded. Third, by means of the Mickens' technique of subequations, a new class of implicit NSFD schemes are constructed for the full convection–diffusion–reaction equations. It is shown that, under certain parameters set, these NSFD schemes are capable of preserving the non‐negativity and boundedness of the analytical solutions. Finally, some numerical simulations are provided to verify the validity of our analytical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1288–1309, 2015     

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.     

A predator-prey model with Beddington-DeAngelis functional response: a non-standard finite-difference method

In this paper, we transform a continuous-time predator-prey model with Beddington–DeAngelis functional response into a discrete-time model by nonstandard finite difference scheme (NSFD). The NSFD model shows complete dynamic consistency with its continuous counterpart for any step size. However, the discrete model of same continuous system obtained by Euler forward method shows dynamic inconsistency for larger step size. Extensive numerical simulations have been done to compare the dynamics of NSFD system and Euler system. Our analysis reveals that dynamics of NSFD model is independent of the step-size, whereas the dynamics of the standard discrete model completely depends on the step-size and produces spurious dynamics like chaos.     

Exact finite-difference schemes for first order differential equations having three distinct fixed-points

We construct nonstandard finite-difference (NSFD) schemes that provide exact numerical methods for a first-order differential equation having three distinct fixed-points. An explicit, but also nonexact, NSFD scheme is also constructed. It has the feature of preserving the critical properties of the original differential equation such as the positivity of the solutions and the stability behavior of the three fixed-points.     

Numerical Analysis for a Mean-Field Equation for the Ising Model with Glauber Dynamics

1.IntroductionWeconsiderthefollowingmean--fieldequationofmotionforthedynamicIsingmodelonaperiodiclatticeA:whereAdenotesthelatticeofZdwithNdsitesdefinedbyA:~{a:a=Zaie',i=1alEZ,15al5N}with{e'}beingthestandardunitvectorsofZd.WesaythatAisad-dimensionallattice.WedenotebyVAtheNddimensionalspaceoflatticevectorsv=(v.).6A*satisfyingv.+Nei=va'Hereu~(u.)..AandbadenotestheexpectationdrofthespinatsiteaofthelatticeandA*isdefinedby{a:a~Za.e',alEZ}.i=1TheNdxNdsymmetricmatrixAisdefinedby3forvEVAF'o…     

Nonstandard finite difference method revisited and application to the Ebola virus disease transmission dynamics

We provide effective and practical guidelines on the choice of the complex denominator function of the discrete derivative as well as on the choice of the nonlocal approximation of nonlinear terms in the construction of nonstandard finite difference (NSFD) schemes. Firstly, we construct nonstandard one-stage and two-stage theta methods for a general dynamical system defined by a system of autonomous ordinary differential equations. We provide a sharp condition, which captures the dynamics of the continuous model. We discuss at length how this condition is pivotal in the construction of the complex denominator function. We show that the nonstandard theta methods are elementary stable in the sense that they have exactly the same fixed-points as the continuous model and they preserve their stability, irrespective of the value of the step size. For more complex dynamical systems that are dissipative, we identify a class of nonstandard theta methods that replicate this property. We apply the first part by considering a dynamical system that models the Ebola Virus Disease (EVD). The formulation of the model involves both the fast/direct and slow/indirect transmission routes. Using the specific structure of the EVD model, we show that, apart from the guidelines in the first part, the nonlocal approximation of nonlinear terms is guided by the productive-destructive structure of the model, whereas the choice of the denominator function is based on the conservation laws and the sub-equations that are associated with the model. We construct a NSFD scheme that is dynamically consistent with respect to the properties of the continuous model such as: positivity and boundedness of solutions; local and/or global asymptotic stability of disease-free and endemic equilibrium points; dependence of the severity of the infection on self-protection measures. Throughout the paper, we provide numerical simulations that support the theory.     

Parallel difference schemes with interface extrapolation terms for quasi-linear parabolic systems

In this paper some new parallel difference schemes with interface extrapolation terms for a quasi-linear parabolic system of equations are constructed. Two types of time extrapolations are proposed to give the interface values on the interface of sub-domains or the values adjacent to the interface points, so that the unconditional stable parallel schemes with the second accuracy are formed. Without assuming heuristically that the original boundary value problem has the unique smooth vector solution, the existence and uniqueness of the discrete vector solutions of the parallel difference schemes constructed are proved. Moreover the unconditional stability of the parallel difference schemes is justified in the sense of the continuous dependence of the discrete vector solution of the schemes on the discrete known data of the original problems in the discrete W2(2,1) (Q△) norms. Finally the convergence of the discrete vector solutions of the parallel difference schemes with interface extrapolation terms to the unique generalized solution of the original quasi-linear parabolic problem is proved. Numerical results are presented to show the good performance of the parallel schemes, including the unconditional stability, the second accuracy and the high parallelism.     

Global stability and a non-standard finite difference scheme for a diffusion driven HBV model with capsids

A diffusion driven model for hepatitis B virus (HBV) infection, taking into account the spatial mobility of both the HBV and the HBV DNA-containing capsids is presented. The global stability for the continuous model is discussed in terms of the basic reproduction number. The analysis is further carried out on a discretized version of the model. Since the standard finite difference (SFD) approximation could potentially lead to numerical instability, it has to be restricted or eliminated through dynamic consistency. The latter is accomplished by using a non-standard finite difference (NSFD) scheme and the global stability properties of the discretized model are studied. The results are numerically illustrated for the dynamics and stability of the various populations in addition to demonstrating the advantages of the usage of NSFD method over the SFD scheme.     

Discretization of an eco-epidemiological model and its dynamic consistency

In this paper, we discretize a continuous-time eco-epidemiological model by non-standard finite difference (NSFD) scheme as well as standard Euler forward scheme. Dynamical properties of both the systems are explored and compared with their continuous-time model. We show that the solution of NSFD system remains positive for all positive initial values. Fixed points and their local stability properties are shown to be identical with that of the continuous model, indicating its dynamic consistency. Dynamics of the Euler model, however, depend on the step–size and therefore dynamically inconsistent. Solutions in this latter method may be negative and allows numerical instabilities, leading to chaos. Extensive numerical simulations have been performed to validate the theoretical results.     

Stability properties of a nonstandard finite difference scheme for a hantavirus epidemic model

This paper studies the stability properties of a nonstandard finite difference (NSFD) scheme used to simulate the dynamics of a mouse population model in hantavirus epidemics. It is shown that this difference scheme and the underlying system of differential equations have the same dynamics. The proof uses the fact that the total population obeys the logistic equation, as well as techniques from calculus, graphical analysis, and dynamical systems.     

Numerical treatment for nonlinear Brusselator chemical model

Nowadays, numerical models have great importance in every field of science, especially for solving the nonlinear differential equations, partial differential equations, biochemical reactions, etc. The total time evolution of the reactant species which interacts with other species is simulated by the Runge-Kutta of order four (RK4) and by Non-Standard finite difference (NSFD) method. A NSFD model has been constructed for the biochemical reaction problem and numerical experiments are performed for different values of discretization parameter h. The results are compared with the well-known numerical scheme, i.e. RK4. The developed scheme NSFD gives better results than RK4.     

Further results on exponential stability of discrete‐time BAM neural networks with time‐varying delays

This paper is concerned with the exponential stability for the discrete‐time bidirectional associative memory neural networks with time‐varying delays. Based on Lyapunov stability theory, some novel delay‐dependent sufficient conditions are obtained to guarantee the globally exponential stability of the addressed neural networks. In order to obtain less conservative results, an improved Lyapunov–Krasovskii functional is constructed and the reciprocally convex approach and free‐weighting matrix method are employed to give the upper bound of the difference of the Lyapunov–Krasovskii functional. Several numerical examples are provided to illustrate the effectiveness of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.     

Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes

In this paper, numerical solution of the Burgers–Huxley (BH) equation is presented based on the nonstandard finite difference (NSFD) scheme. At first, two exact finite difference schemes for BH equation obtained. Moreover an NSFD scheme is presented for this equation. The positivity, boundedness and local truncation error of the scheme are discussed. Finally, the numerical results of the proposed method with those of some available methods compared.     

一类具有扩散和时滞的离散复合种群模型的Hopf分岔

本文讨论了生物上一类有时滞和扩散(迁移)的离散复合种群模型．利用离散系统相关结果分析了该模型的正不动点的类型及稳定性,并用中心流形方法对原系统降维从而讨论了它的Hopf分岔问题以及扩散和时滞对种群生态学的意义．     

Some temporal second order difference schemes for fractional wave equations

In this article, motivated by Alikhanov's new work (Alikhanov, J Comput Phys 280 (2015), 424–438), some difference schemes are proposed for both one‐dimensional and two‐dimensional time‐fractional wave equations. The obtained schemes can achieve second‐order numerical accuracy both in time and in space. The unconditional convergence and stability of these schemes in the discrete H¹‐norm are proved by the discrete energy method. The spatial compact difference schemes with the results on the convergence and stability are also presented. In addition, the three‐dimensional problem is briefly mentioned. Numerical examples illustrate the efficiency of the proposed schemes. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 970–1001, 2016     

Nonstandard finite difference methods: recent trends and further developments

In this paper, we review many recent developments and further applications of nonstandard finite difference (NSFD) methods encountered in the past decade. In particular, it is a follow up article of the one published in 2005 [K.C. Patidar, On the use of non-standard finite difference methods, J. Differ. Equ. Appl. 11 (2005), pp. 735–758]. It also includes those research contributions in this field that are very significant and published prior to the above article but were not included in the above paper simply because we did not have access to them when we wrote the above article. We also give a detailed account on various definitions/notions of NSFD methods appeared in the literature in past two decades. All contributions are listed chronologically except that in some instances we have grouped certain works to show connectivity in those fields. While categorizing these research contributions, we considered a number of different application areas. Moreover, due to space limitations, firstly, we have not included all works that used NSFD methodology but certainly important contributions are given due consideration, and secondly, we have only included the salient features of the proposed numerical schemes for many of these contributions and ignored other contents where the involved approaches sound fairly standard thus implying why these methods are known robust. Of course, the cases when there is a specific variation, in the theoretical analysis of these NSFD schemes, are highlighted with more details.