Nonstandard Finite Difference Scheme Research Articles

A note on a positivity preserving hyperbolic NSFD scheme for heat transfer

We construct a new nonstandard finite difference (NSFD) scheme for a hyperbolic partial differential equation (PDE) modelling heat transfer. This discretization satisfies a positivity condition for its solutions and is consistent in its general features with the experimental data. A brief discussion is also provided as to how the “additional” initial condition, , should be implemented on the computational grid. Further, we examine the details of the mathematical structure of the scheme and their implications for systems involving heat transfer.

Dynamic analysis of a predator-prey model with Holling type-II functional response and prey refuge by using a NSFD scheme

This study proposes and analyzes a nonstandard finite difference (NSFD) scheme to retain the fundamental qualitative aspects of a predator-prey interaction with a Holling type-II functional response and prey refuge. The existence and stability of fixed points are examined. A few numerical examples are presented to back up the theoretical findings. The results show that the numerical simulations match well with the theoretical outcomes. Moreover, the model experiences transcritical, period-doubling, and Neimark-Sacker bifurcation. Furthermore, numerical simulations show that standard numerical methods like the Euler method and the classical RK4 method are not dynamically consistent with the continuous model. As a result, they do not accurately reflect the continuous model’s behavior. The proposed NSFD scheme, on the other hand, is shown to be suitable and adequate for solving the continuous model.

Construction a distributed order smoking model and its nonstandard finite difference discretization

<abstract><p>Smoking is currently one of the most important health problems in the world and increases the risk of developing diseases. For these reasons, it is important to determine the effects of smoking on humans. In this paper, we discuss a new system of distributed order fractional differential equations of the smoking model. With the use of distributed order fractional differential equations, it is possible to solve both ordinary and fractional-order equations. We can make these solutions with the density function included in the definition of the distributed order fractional differential equation. We construct the Nonstandard Finite Difference (NSFD) schemes to obtain numerical solutions of this model. Positivity solutions are preserved under positive initial conditions with this discretization method. Also, since NSFD schemes can preserve all the properties of the continuous models for any discretization parameter, the method is successful in dynamical consistency. We use the Schur-Cohn criteria for stability analysis of the discretized model. With the solutions obtained, we can understand the effects of smoking on people in a short time, even in different situations. Thus, by knowing these effects in advance, potential health problems can be predicted, and life risks can be minimized according to these predictions.</p></abstract>

Numerical solutions of nonlinear Burgers‒Huxley equation through the Richtmyer type nonstandard finite difference method

The Burger‒Huxley equation as a well-known nonlinear physical model is studied numerically in the present paper. In this respect, the nonstandard finite difference (NSFD) scheme in company with the Richtmyer’s (3, 1, 1) implicit formula is formally adopted to accomplish this goal. Moreover, the stability, convergence, and consistency analyses of nonstandard finite difference schemes are investigated systematically. Several case studies with comparisons are provided, confirming that the current numerical scheme is capable of resulting in highly accurate approximations.

A nonstandard finite difference scheme for the SVICDR model to predict COVID-19 dynamics.

In the context of 2019 coronavirus disease (COVID-19), considerable attention has been paid to mathematical models for predicting country- or region-specific future pandemic developments. In this work, we developed an SVICDR model that includes a susceptible, an all-or-nothing vaccinated, an infected, an intensive care, a deceased, and a recovered compartment. It is based on the susceptible-infectious-recovered (SIR) model of Kermack and McKendrick, which is based on ordinary differential equations (ODEs). The main objective is to show the impact of parameter boundary modifications on the predicted incidence rate, taking into account recent data on Germany in the pandemic, an exponential increasing vaccination rate in the considered time window and trigonometric contact and quarantine rate functions. For the numerical solution of the ODE systems a model-specific non-standard finite difference (NSFD) scheme is designed, that preserves the positivity of solutions and yields the correct asymptotic behaviour.

Uniformly Convergent Scheme for Singularly Perturbed Space Delay Parabolic Differential Equation with Discontinuous Convection Coefficient and Source Term

A singularly perturbed delay parabolic problem of convection‐diffusion type with a discontinuous convection coefficient and source term is examined. In the problem, strong interior layers and weak boundary layers are exhibited due to a large delay in the spatial variable and discontinuity of convection coefficient and source. The problem is discretized by a nonstandard finite difference scheme in the spatial variable and for the time derivative, we used the Crank–Nicolson scheme. To enhance the order of convergence of the spatial variable, the Richardson extrapolation technique is applied. The error analysis of the proposed scheme was carried out and proved that the scheme is uniformly convergent of second order in both spatial and temporal variables. Numerical experiments are performed to verify the theoretical estimates.

A finite difference scheme to solve a fractional order epidemic model of computer virus

<abstract><p>In this article, an analytical and numerical analysis of a computer virus epidemic model is presented. To more thoroughly examine the dynamics of the virus, the classical model is transformed into a fractional order model. The Caputo differential operator is applied to achieve this. The Jacobian approach is employed to investigate the model's stability. To investigate the model's numerical solution, a hybridized numerical scheme called the Grunwald Letnikov nonstandard finite difference (GL-NSFD) scheme is created. Some essential characteristics of the population model are scrutinized, including positivity boundedness and scheme stability. The aforementioned features are validated using test cases and computer simulations. The mathematical graphs are all detailed. It is also investigated how the fundamental reproduction number $ \mathfrak{R}_0 $ functions in stability analysis and illness dynamics.</p></abstract>

Numerical approximations of stochastic Gray-Scott model with two novel schemes

<abstract><p>This article deals with coupled nonlinear stochastic partial differential equations. It is a reaction-diffusion system, known as the stochastic Gray-Scott model. The numerical approximation of the stochastic Gray-Scott model is discussed with the proposed stochastic forward Euler (SFE) scheme and the proposed stochastic non-standard finite difference (NSFD) scheme. Both schemes are consistent with the given system of equations. The linear stability analysis is discussed. The proposed SFE scheme is conditionally stable and the proposed stochastic NSFD is unconditionally stable. The convergence of the schemes is also discussed in the mean square sense. The simulations of the numerical solution have been obtained by using the MATLAB package for the various values of the parameters. The effects of randomness are discussed. Regarding the graphical behavior of the stochastic Gray-Scott model, self-replicating behavior is observed.</p></abstract>

A Convergence-Preserving Non-Standard Finite Difference Scheme for the Solutions of Singular Lane-Emden Equations

A Convergence-Preserving Non-Standard Finite Difference Scheme for the Solutions of Singular Lane-Emden Equations

Structure Preserving Algorithm for Fractional Order Mathematical Model of COVID-19

In this article, a brief biological structure and some basic properties of COVID-19 are described. A classical integer order model is modified and converted into a fractional order model with as order of the fractional derivative. Moreover, a valued structure preserving the numerical design, coined as Grunwald–Letnikov non-standard finite difference scheme, is developed for the fractional COVID-19 model. Taking into account the importance of the positivity and boundedness of the state variables, some productive results have been proved to ensure these essential features. Stability of the model at a corona free and a corona existing equilibrium points is investigated on the basis of Eigen values. The Routh–Hurwitz criterion is applied for the local stability analysis. An appropriate example with fitted and estimated set of parametric values is presented for the simulations. Graphical solutions are displayed for the chosen values of (fractional order of the derivatives). The role of quarantined policy is also determined gradually to highlight its significance and relevancy in controlling infectious diseases. In the end, outcomes of the study are presented.

Stability analysis and Neimark-Sacker bifurcation of a nonstandard finite difference scheme for Lotka-Volterra prey-predator model

Stability analysis and Neimark-Sacker bifurcation of a nonstandard finite difference scheme for Lotka-Volterra prey-predator model

Design, Analysis and Comparison of a Nonstandard Computational Method for the Solution of a General Stochastic Fractional Epidemic Model

Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the model at equilibrium points is investigated by applying the Jacobian matrix technique. The contribution of the basic reproduction number, R0, in the infection dynamics and stability analysis is elucidated. The results indicate that the given system is locally asymptotically stable at the disease-free steady-state solution when R0<1. A similar result is obtained for the endemic equilibrium when R0>1. The underlying system shows global stability at both steady states. The fractional-order system is converted into a stochastic model. For a more realistic study of the disease dynamics, the non-parametric perturbation version of the stochastic epidemic model is developed and studied numerically. The general stochastic fractional Euler method, Runge–Kutta method, and a proposed numerical method are applied to solve the model. The standard techniques fail to preserve the positivity property of the continuous system. Meanwhile, the proposed stochastic fractional nonstandard finite-difference method preserves the positivity. For the boundedness of the nonstandard finite-difference scheme, a result is established. All the analytical results are verified by numerical simulations. A comparison of the numerical techniques is carried out graphically. The conclusions of the study are discussed as a closing note.

A Reliable numerical scheme for a computer virus propagation model

n this paper, we develop the Mickens’ methodology to construct a dynamically consistent Non-Standard FiniteDifference (NSFD) scheme for a recognized computer virus propagation model. It is proved that the constructed NSFD scheme correctly preserves essential mathematical features of the continuous-time model, which are positivity, boundedness and asymptotic stability. As an important consequence, we obtain an effective numerical scheme that has the ability to provide reliable approximations, meanwhile, some typical standard finite difference schemes fail to preserve the essential properties of the computer virus propagation model; hence, they can generate numerical approximations which are not only negative but also unstable. Finally, a set of numerical experiments is performed to support the theoretical results as well as to demonstrate the advantage of the NSFD scheme over standard ones. As we expected, there is a good agreement between the numerical results and theoretical assertions.

Numerical and semi-numerical solutions of a modified Thévenin model for calculating terminal voltage of battery cells

A new Thévenin-type model for evaluating the terminal voltage of rechargeable batteries is developed by adding a nonlinear capacitor element in the circuit. Furthermore, a powerful variant of the improved differential transform method and the nonstandard finite difference schemes are proposed to properly treat the mathematical model. Two separate case studies including a constant and a variable current consumer, with latter representing an inductive electric motor, are carried out. The efficacies of the multistage improved differential transform method, the nonstandard finite difference schemes, and the classical Euler method are compared using a set of appropriate CPU-time and error analyses.

Dynamically consistent nonstandard finite difference schemes for a virus-patch dynamic model

Dynamically consistent nonstandard finite difference schemes for a virus-patch dynamic model

A robust study of a piecewise fractional order COVID-19 mathematical model

In the current manuscript, we deal with the dynamics of a piecewise covid-19 mathematical model with quarantine class and vaccination using SEIQR epidemic model. For this, we discussed the deterministic, stochastic, and fractional forms of the proposed model for different steps. It has a great impact on the infectious disease models and especially for covid-19 because in start the deterministic model played its role but with time due to uncertainty the stochastic model takes place and with long term expansion the use of fractional derivatives are required. The stability of the model is discussed regarding the reproductive number. Using the non-standard finite difference scheme for the numerical solution of the deterministic model and illustrate the obtained results graphically. Further, environmental noises are added to the model for the description of the stochastic model. Then take out the existence and uniqueness of positive solution with extinction for infection. Finally, we utilize a new technique of piecewise differential and integral operators for approximating Caputo-Fabrizio fractional derivative operator for the purpose of constructing of the fractional-order model. Then study the dynamics of the models such as positivity and boundedness of the solutions and local stability analysis. Solved numerically fractional-order model used Newton Polynomial scheme and present the results graphically.

Some Finite Difference Methods to Model Biofilm Growth and Decay: Classical and Non-Standard

The study of biofilm formation is undoubtedly important due to micro-organisms forming a protected mode from the host defense mechanism, which may result in alteration in the host gene transcription and growth rate. A mathematical model of the nonlinear advection–diffusion–reaction equation has been studied for biofilm formation. In this paper, we present two novel non-standard finite difference schemes to obtain an approximate solution to the mathematical model of biofilm formation. One explicit non-standard finite difference scheme is proposed for biomass density equation and one property-conserving scheme for a coupled substrate–biomass system of equations. The nonlinear term in the mathematical model has been handled efficiently. The proposed schemes maintain dynamical consistency (positivity, boundedness, merging of colonies, biofilm annihilation), which is revealed through experimental observation. In order to verify the accuracy and effectiveness of our proposed schemes, we compare our results with those obtained from standard finite difference schemes and earlier known results in the literature. The proposed schemes (NSFD1 and NSFD2) show good performance. The NSFD2 scheme reveals that the processes of biofilm formation and nutritive substrate growth are intricately linked.

Bifurcation analysis of a computer virus propagation model

We propose a mathematical model for investigating the efficacy of Countermeasure Competing (CMC) strategy which is a method for reducing the effect of computer virus attacks. Using the Centre Manifold Theory, we determine conditions under which a subcritical (backward) bifurcation occurs at Basic Reproduction Number $R_{0}=1$. In order to illustrate the theoretical findings, we construct a new Nonstandard Finite Difference Scheme (NSFD) that preserves the bifurcation property at $R_{0}=1$ among other dynamics of the continuous model. Earlier results given by Chen and Carley [The impact of countermeasure propagation on the prevalence of computer viruses, IEEE Trans. Syst., Man, Cybern. B. Cybern. 2004] show that the CMC strategy is effective when the countermeasure propagation rate is higher than the virus spreading rate. Our results reveal that even if this condition is not met, the CMC strategy may still successfully eradicate computer viruses depending on the extent of its effectiveness.

On a weakly L‐stable time integration formula coupled with nonstandard finite difference scheme with application to nonlinear parabolic partial differential equations

In the present paper, we establish an efficient numerical scheme based on weakly L‐stable time integration convergent formula and nonstandard finite difference (NSFD) scheme. We solve Burgers' equation with Dirichlet boundary conditions as well as Neumann boundary conditions. We also solve the Fisher equation. We use Hermite approximation polynomial of order five and backward explicit Taylor's series approximation of order six to derive the numerical integration formula for the initial value problem (IVP) . We combine this method with the NSFD scheme and convert the problem into the system of algebraic equations. We discuss the convergence, truncation error, and stability of the developed method. To demonstrate the efficiency of the developed method, we compare the numerical results with some existing numerical results and exact solutions.

Reliable approximations for a hepatitis B virus model by nonstandard numerical schemes

In this work, we propose and analyze nonstandard finite difference (NSFD) schemes for an improved hepatitis B virus (HBV) model. Dynamical properties of the constructed NSFD schemes are studied rigorously. It is proven that the NSFD schemes preserve positivity, boundedness and asymptotic stability of the HBV model for all finite step sizes. In addition, convergence and error bounds for the NSFD schemes are also investigated. The theoretical results and advantages of the NSFD schemes over some existing standard finite difference ones are supported and illustrated by a set of numerical examples. The numerical results show that some typical standard numerical schemes cannot preserve the dynamics of the continuous model for some given step sizes; consequently, they generate numerical solutions that are not only unstable but also negative. Conversely, the constructed NSFD schemes preserve the essential mathematical features of the continuous model for any finite step size; therefore, they are suitable to simulate the dynamics of the HBV model over long time intervals.