Nonstandard Difference Scheme Research Articles

Exact numerical solutions and high order nonstandard difference schemes for a second order delay differential equation

In this work, we obtain an expression, given in terms of some special functions, for the exact numerical solution of the second order delay differential equation with general initial condition for . Based on this exact solution, we propose a family of nonstandard finite difference methods that combine high order accuracy and efficient computational properties. We also show that the numerical solutions obtained with the new schemes are consistent with asymptotic stability properties of the exact solutions.

On the qualitative study of a discrete-time phytoplankton-zooplankton model under the effects of external toxicity in phytoplankton population

The current manuscript studies a discrete-time phytoplankton-zooplankton model with Holling type-II response. The original model is modified by considering the condition that the phytoplankton population is getting infected with an external toxic substance. To obtain the discrete counterpart from a continuous-time system, Euler's forward method is applied. Moreover, a consistent discrete-time phytoplankton-zooplankton model is obtained by using a nonstandard difference scheme. The boundedness character for every positive solution is discussed, and the local stability of obtained system about each of its fixed points is discussed. The existence of period-doubling bifurcation at a positive equilibrium point is discussed for the discrete system obtained by Euler's forward method. In addition, the comparison of the consistent discrete-time version with its inconsistent counterpart is provided. It is proved that the discrete-time system obtained by using a nonstandard scheme is dynamically consistent as there is no chance for the existence of period-doubling bifurcation in that system. In order to control the period-doubling bifurcation and Neimark-Sacker bifurcation, an improved hybrid control strategy is applied. Finally, we have provided some interesting numerical examples to explain our theoretical results.

Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model

This research manifesto has a comprehensive discussion of the global dynamics of an achievable discrete-time two predators and one prey Lotka–Volterra model in three dimensions, i.e., in the space R3. In some assertive parametric circumstances, the discrete-time model has eight equilibrium points among which one is a special or unique positive equilibrium point. We have also investigated the local and global behavior of equilibrium points of an achievable three-dimensional discrete-time two predators and one prey Lotka–Volterra model. The conversion of a continuous-type model into its discrete counterpart model has been completed by adopting a dynamically consistent nonstandard difference scheme with the end goal that the equilibrium points are conserved in twin cases. The difficulty lies in how to find all fixed points O,P,Q,R,S,T,U,V and the Jacobian matrix and its characteristic polynomial at the unique positive fixed point. For that purpose, we use Mathematica software to find the equilibrium points and all of the Jacobian matrices at those equilibrium points. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of the obtained system about all of its equilibrium points. The discrete Lotka–Volterra model in three dimensions is given by system (3), where parameters α,β,γ,δ,ζ,η,μ,ε,υ,ρ,σ,ω∈R+ and initial conditions x0,y0,z0 are positive real numbers. Additionally, the rate of convergence of a solution that converges to a unique positive equilibrium point is discussed. To represent theoretical perceptions, some numerical debates are introduced, including phase portraits.

On the qualitative study of a two-trophic plant-herbivore model.

The coexistence of plant-herbivore populations in an ecological system is a fundamental topic of research in mathematical ecology. Plant-herbivore interactions are often described by using discrete-time models in the case of non-overlapping generations: such generations have some specific time interval of life and their old generations are replaced by new generations after some regular interval of time. Keeping in mind the dynamical reliability of continuous-time models we presented two discrete-time plant-herbivore models. Mainly, by applying Euler's forward method a discrete-time plant-herbivore model is obtained from a continuous-time plant-herbivore model. In addition, a dynamically consistent discrete-time plant-herbivore model is obtained by applying a nonstandard difference scheme. Moreover, local stability is discussed and the existence of bifurcation about positive equilibrium is shown under some mathematical conditions. To control bifurcation and chaos, a modified hybrid technique is developed. Finally, to support our theocratical results and to show the dynamical reliability of the nonstandard difference scheme some numerical examples are provided.

A Dynamically Consistent Nonstandard Difference Scheme for a Discrete-Time Immunogenic Tumors Model.

This manuscript deals with the qualitative study of certain properties of an immunogenic tumors model. Mainly, we obtain a dynamically consistent discrete-time immunogenic tumors model using a nonstandard difference scheme. The existence of fixed points and their stability are discussed. It is shown that a continuous system experiences Hopf bifurcation at one and only one positive fixed point, whereas its discrete-time counterpart experiences Neimark–Sacker bifurcation at one and only one positive fixed point. It is shown that there is no chance of period-doubling bifurcation in our discrete-time system. Additionally, numerical simulations are carried out in support of our theoretical discussion.

Qualitative behavior of a two‐dimensional discrete‐time prey–predator model

In this article, we discuss the qualitative behavior of a two-dimensional discrete-time prey–predator model. This system is the result of the application of a nonstandard difference scheme to a system of differential equations for a prey–predator model including intraspecific competition of prey population. In particular, we evaluate the fixed points of the system and study their local asymptotic stability. We also prove the existence of a Neimark–Sacker bifurcation.

Qualitative analysis of a discrete-time phytoplankton\u2013zooplankton model with Holling type-II response and toxicity

The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton–zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark–Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper.

Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay

We propose a mathematical model based on a set of delay differential equations that describe intracellular HIV infection. The model includes three different subpopulations of cells and the HIV virus. The mathematical model is formulated in such a way that takes into account the time between viral entry into a target cell and the production of new virions. We study the local stability of the infection-free and endemic equilibrium states. Moreover, by using a suitable Lyapunov functional and the LaSalle invariant principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable. In addition, we designed a non-standard difference scheme that preserves some relevant properties of the continuous mathematical model.

Bifurcation analysis of a discrete-time four-dimensional cubic autocatalator chemical reaction model with coupling through uncatalysed reactant

In this manuscript, we discuss a four-dimensional cubic autocatalator chemical reaction model in continuous form. We investigate the existence of one and only positive fixed point and then we have obtained some parametric conditions for local stability of continuous system by using Routh-Hurwitz stability criteria. Moreover, we discretize the four-dimensional continuous cubic autocatalator chemical reaction model by using Euler’s forward method and then by using a nonstandard difference scheme we obtained a consistent discrete-time counterpart of four-dimensional cubic autocatalator chemical reaction model. Parametric conditions for local asymptotic stability of one and only positive fixed point of obtained system are also discussed. It is shown that the obtained system experiences the Neimark-Sacker bifurcation at one and only positive fixed point by using a general standard for Neimark-Sacker bifurcation. The discrete-time counterpart of genuine four-dimensional system displays chaotic dynamics at different standards of bifurcation parameter. Furthermore, the control of Neimark-Sacker bifurcation and chaos is also deliberated by using a generalized hybrid control scheme, which is based on parameter perturbation and feedback control. Finally, some numerical examples are given to strengthen our theoretical results.

A cubic autocatalator chemical reaction model with limit cycle analysis and consistency preserving discretization

This article deals with the study of some qualitative properties of a cubic autocatalator chemical reaction model. Particularly, we obtain a dynamically consistent cubic autocatalator discrete-time model by applying a nonstandard difference scheme. Analysis of the existence of equilibria and their stability is carried out. It is proved that a continuous system undergoes the Hopf bifurcation at its interior equilibrium, whereas the discrete-time version undergoes Neimark-Sacker bifurcation at its interior fixed point. Moreover, numerical simulation is provided to strengthen our theoretical discussion.

A novel chaos control strategy for discrete-time Brusselator models

This article deals with the dynamical analysis of discrete-time Brusselator models. Euler’s forward and nonstandard difference schemes are implemented for discretization of Brusselator system. We investigate the local dynamics related to equilibria of both discrete-time models. Furthermore, with the help of bifurcation theory and center manifold theorem, explicit parametric conditions for directions and existence of flip and Hopf bifurcations are investigated. A novel chaos control method is implemented in order to control chaos in discrete-time Brusselator models under the influence of flip and Hopf bifurcations. Numerical simulations are provided to illustrate theoretical discussion and effectiveness of newly introduced chaos control strategy.

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.

Two non-standard finite difference schemes for the Timoshenko beam

In this paper, we derive two non-standard finite difference schemes for the Timoshenko beam problem. The schemes are uniform with respect to the thickness small parameter of the beam, thus, no locking phenomenon occurs. Numerical results are presented. Key words: Timoshenko beam, non-standard difference scheme, locking phenomenon

Nonstandard numerical schemes for modeling a 2-DOF serial robot with rotational spring-damper-actuators

In this paper we construct and develop different nonstandard finite difference schemes for a 2 degree of freedom serial robot with rotational spring–damper–actuators (RSDA). The mathematical model of the system is developed using Gibbs–Appell (G–A) equation of motion and the resulting symbolic expressions are written in configuration-space form allowing the exploitation of some characteristics related to centrifugal and coriolis forces such that nonlocal approximations can be applied to the quadratic and product joint velocity terms. To obtain numerical solutions of these highly nonlinear and coupled differential equations is not a straightforward task. Here, we show that nonstandard finite difference schemes increase the numerical stability region for the time step size. Our schemes can be divided into two approaches: the first is when denominators of the discrete derivatives are defined using nontraditional functions of the time step size in order to ensure that the fixed (equilibrium) points of the resulting discrete system has the same stability properties as those of the original system. In the second approach, the nonlinear terms are replaced by nonlocal discrete representations. Numerical comparisons of these numerical schemes are performed with Runge–Kutta-type methods in order to observe the advantages of the nonstandard difference schemes. Numerical simulations of the dynamics of robotic systems with these nonstandard numerical schemes give more reliable and robust results. Copyright © 2009 John Wiley & Sons, Ltd.

A Nonstandard Dynamically Consistent Numerical Scheme Applied to Obesity Dynamics

The obesity epidemic is considered a health concern of paramount importance in modern society. In this work, a nonstandard finite difference scheme has been developed with the aim to solve numerically a mathematical model for obesity population dynamics. This interacting population model represented as a system of coupled nonlinear ordinary differential equations is used to analyze, understand, and predict the dynamics of obesity populations. The construction of the proposed discrete scheme is developed such that it is dynamically consistent with the original differential equations model. Since the total population in this mathematical model is assumed constant, the proposed scheme has been constructed to satisfy the associated conservation law and positivity condition. Numerical comparisons between the competitive nonstandard scheme developed here and Euler′s method show the effectiveness of the proposed nonstandard numerical scheme. Numerical examples show that the nonstandard difference scheme methodology is a good option to solve numerically different mathematical models where essential properties of the populations need to be satisfied in order to simulate the real world.

Numerical simulation for a Duffing oscillator driven by colored noise using nonstandard difference scheme

The present paper is devoted to the numerical solution of the Fokker–Planck (FP) equation associated to the Duffing oscillator driven by colored noise. We propose an improved discretization of the standard FP operator-splitting method which renders the scheme unconditionally stable. This scheme is successfully applied to explore the influence of the colored noise on the probability density.

Explicit finite-difference time-domain method with perfectly matched layers for transient low-frequency eddy-current analysis

The finite-difference time-domain (FDTD) method is a robust algorithm for the solution of high-frequency electromagnetic problems, but, unfortunately, the method has no counterpart in eddy-current analysis. Hence, the extension of FDTD to power frequencies is an attractive theoretical problem that could give rise to an entirely new methodology for low-frequency electromagnetics. In this paper, we introduce a general explicit FDTD algorithm for transient eddy-current problems. We perform a theoretical investigation of a nonstandard difference scheme, including the issues of stability and consistency. The study provides a class of explicit methods with varying properties and computational complexity. From these, we chose the DuFort-Frankel algorithm for its simplicity and efficiency. However, the most intricate issue for the application of an explicit scheme is the solution of the open boundary problem. Unlike conventional integral equation or forced truncation approaches, the free-space problem is successfully treated by using a specially designed perfectly matched layer (PML) for eddy currents. The explicit scheme within the conductors is, finally, efficiently combined with the free space-PML equations via the interface conditions.

Non-standard difference schemes for singular perturbation problems revisited

Non-standard finite difference techniques for solving various linear two-point singularly perturbed boundary value problems are considered. They consist of replacing the original differential equation by a related differential equation with a small deviating argument. This related problem is then discretized using finite differences. This discretized problem is solved using standard matrix methods. The accuracy and robustness of these numerical schemes are then carefully investigated, and compared to standard centered finite differences. In particular, earlier claims for the superiority of such schemes over standard finite differences are shown to be invalid.

Asymptotic properties of solutions to two discrete airy equations*

We construct two finite difference models for the Airy differential equation. In one model, the form of the complete asymptotic representation of the solution can be found. However, this is not the case for the second model which is based on the use of a nonstandard difference scheme. This scheme leads to a second-order, linear difference equation that is not of a form for which the theorems of Poincaré and Perron can be directly applied to obtain the asymptotic behavior of the solutions.