Nonstandard Finite Difference Scheme Research Articles

Wavelet-based approximation with nonstandard finite difference scheme for singularly perturbed partial integrodifferential equations

Wavelet-based approximation with nonstandard finite difference scheme for singularly perturbed partial integrodifferential equations

A convergence-preserving non-standard finite difference scheme for the solutions of singular Lane-Emden equations

The derivation of numerical schemes for the solution of Lane-Emden equations requires meticulous consideration because they are highly nonlinear in nature; have singularity behaviors at the origin and in some cases, their exact solutions are known for only a few parameter ranges. This explains the reason for the failure of some existing methods in approximating such problems. Thus, in this research, a convergence-preserving Non-Standard Finite Difference Scheme (NSFDS) shall be derived for the solution of Lane-Emden equations. The main idea in this work is the approximation of both the linear and nonlinear terms non-locally and the renormalization/reformulation of the numerator and denominator functions of the Lane-Emden equations. The need for this approach came up due to some setbacks of existing methods where the exact solutions’ qualitative properties are not routinely transferred to the approximate solutions. The analysis of some properties of the NSFDS like convergence, dynamical consistency, monotone dependence on the initial value, and monotonicity of solutions as well as perturbation solution analysis shall be carried out. The NSFDS was then employed in solving some classes of singular initial value problems of the Lane-Emden type and the results obtained show that the proposed method is efficient and computationally cheap.

The Stability Analysis and Transmission Dynamics of the SIR Model with Nonlinear Recovery and Incidence Rates

In the present paper, the SIR model with nonlinear recovery and Monod type equation as incidence rates is proposed and analyzed. The expression for basic reproduction number is obtained which plays a main role in the stability of disease-free and endemic equilibria. The nonstandard finite difference (NSFD) scheme is constructed for the model and the denominator function is chosen such that the suggested scheme ensures solutions boundedness. It is shown that the NSFD scheme does not depend on the step size and gives better results in all respects. To prove the local stability of disease-free equilibrium point, the Jacobean method is used; however, Schur–Cohn conditions are applied to discuss the local stability of the endemic equilibrium point for the discrete NSFD scheme. The Enatsu criterion and Lyapunov function are employed to prove the global stability of disease-free and endemic equilibria. Numerical simulations are also presented to discuss the advantages of NSFD scheme as well as to strengthen the theoretical results. Numerical simulations specify that the NSFD scheme preserves the important properties of the continuous model. Consequently, they can produce estimates which are entirely according to the solutions of the model.

A construction of nonstandard finite difference scheme to fractional differential system

A construction of nonstandard finite difference scheme to fractional differential system

Approximate Solutions for a Class of Predator–Prey Systems with Nonstandard Finite Difference Schemes

In this paper, we construct new nonstandard finite difference schemes to approximate a set of positive solutions for the predator–prey model, which contains different functional responses. The organization of the denominator of the discrete derivative and nonlocal approximations of nonlinear terms are employed to design the new schemes. The approach results in significant qualitative improvements in how the numerical solution behaves. We establish that the proposed nonstandard finite difference methods are elementary stable and satisfy the positivity requirement. In addition, the instances of applying PESN methods to some predator–prey systems using the Beddington–DeAngelis and Nicholson–Bailey functional responses are provided here. Finally, some numerical comparisons are presented to illustrate our findings. Our results indicate that the proposed methods are very suitable for the symmetric model of predator–prey.

MATHEMATICAL AND NUMERICAL ANALYSIS OF SIQR EPIDEMIC MODEL OF MEASLES DISEASE DYNAMICS

Measles is a highly transmissible disease in children around the world. According to the World Health Organization (WHO), 73% of deaths of children were due to measles in 2018. This study describes the physical solution of the SIQR model for measles spread under the effect of natural delay amongst different compartments. By three different numerical techniques, the efficacies of solutions of the underlying system have been compared and a clear preference of nonstandard finite-difference (NSFD) scheme over the rest has been established. It has also been observed, on principle, that the NSFD formulation recovers all the essential traits of a continuous model namely the boundedness, positivity and stability of equilibriums of populations. The numerical results have also been supported by a very strong classical analysis of the model where the existence of a solution vector in explicit subsets of the function spaces has been guaranteed which leads to optimization of fixed-point methods.

DYNAMICAL BEHAVIOURS OF A DISCRETIZED MODEL WITH MICHAELIS-MENTEN HARVESTING RATE

In this paper, we introduced nonstandard finite difference scheme (NSFD) for solving the continuos model with Michaelis-Menten harvesting rate. We have seen that the proposed scheme preserve local stability and positivity. Stability analysis of each fixed point of the discrete time model has been proven. Also, numerical comparisons were made between the nonstandard finite difference method and the other methods.

To study the transmission dynamic of SARS-CoV-2 using nonlinear saturated incidence rate

In this work, we construct a new SARS-CoV-2 mathematical model of SQIR type. The considered model has four compartments as susceptible S, quarantine Q, infected I and recovered R. Here saturated nonlinear incidence rate is used for the transmission of the disease. We formulate our model first and then the disease-free and endemic equilibrium (EE) are calculated. Further, the basic reproduction number is computed via the next generation matrix method. Also on using the idea of Dulac function, the global stability for the proposed model is discussed. By using the Routh–Hurwitz criteria, local stability is investigated. Through nonstandard finite difference (NSFD) scheme, numerical simulations are performed. Keeping in mind the significant importance of fractional calculus in recent time, the considered model is also investigated under fractional order derivative in Caputo sense. Finally, numerical interpretation of the model by using various fractional order derivatives are provided. For fractional order model, we utilize fractional order NSFD method. Comparison with some real data is also given.

Convergence Analysis and Approximate Optimal Temporal Step Sizes for Some Finite Difference Methods Discretising Fisher's Equation

In this study, we obtain a numerical solution for Fisher's equation using a numerical experiment with three different cases. The three cases correspond to different coefficients for the reaction term. We use three numerical methods namely; Forward-Time Central Space (FTCS) scheme, a Nonstandard Finite Difference (NSFD) scheme, and the Explicit Exponential Finite Difference (EEFD) scheme. We first study the properties of the schemes such as positivity, boundedness, and stability and obtain convergence estimates. We then obtain values of L1 and L∞ errors in order to obtain an estimate of the optimal time step size at a given value of spatial step size. We determine if the optimal time step size is influenced by the choice of the numerical methods or the coefficient of reaction term used. Finally, we compute the rate of convergence in time using L1 and L∞ errors for all three methods for the three cases.

Nonstandard Discretization and Stability Analysis of a novel type Malaria-Ross Model

Malaria is still a deadly disease in most developing countries. In order to prevent this and many other diseases in all countries, it is necessary to understand the dynamics of the disease well. For this reason, in this study, a new type of Malaria-Ross equation, Distributed order, is discussed. In this new type, the dynamics of the disease can be understood better and quicker in different situations with the density function included in such equations. Numerical discretization of this model is done with the help of a nonstandard finite difference scheme. Afterward, stability analyses of the equilibrium points obtained from the model that were performed. At the same time, comparisons were made with other numerical methods. Finally, the findings are expressed with graphs and tables.

DETERMINISTIC AND STOCHASTIC ANALYSIS OF A COVID-19 SPREAD MODEL

This paper deals with the global dynamics of deterministic-stochastic COVID-19 mathematical model with quarantine class and incorporating a preventive vaccination. Lyapunov functions are utilized for the global stability of disease free equilibrium point and the graph theoretic method is used for the construction of Lyapunov function for positive equilibrium point. The stability of model is discussed regarding the reproductive number. Utilizing the non-standard finite difference scheme for the numerical solution of the deterministic model, the obtained results are shown graphically. Further, environmental noises are added to the model for description of stochastic model. Then we take out the existence and uniqueness of positive solution with extinction for infection. Finally, we solve numerically the stochastic model using Newton Polynomial scheme and present the results graphically.

On the symmetrization and composition of nonstandard finite difference schemes as an alternative to Richardson's extrapolation

From the classical explicit Euler scheme of first order, nonstandard finite difference (NSFD) schemes were envisioned to mimic the essential properties of the governing differential equation model for every time step-size. In the context of compartmental epidemiological models, these properties are generally concerned with the positivity of subpopulations, conservation laws (dynamics of the total population), and stability. However, for autonomous systems, the symmetry (self-adjoint) condition is not preserved. Compartmental epidemiological models are Poisson systems, so methods from geometric numerical integration should be applicable. It is found that symmetrization of NSFD schemes does not respect positivity for every step-size, though other characteristics are maintained and second order is reached. Composition through the Lie formalism can then be applied to obtain higher-order schemes. This is a more efficient and consistent alternative to Richardson's extrapolation, which has often been used to go beyond order one.

On the choice of denominator functions and convergence of NSFD schemes for a class of nonlinear SBVPs

We propose two novel non-standard finite difference (NSFD) schemes for a class of non-linear singular boundary value problems (SBVPs). One of the basic idea of NSFD schemes is that the step size Δt is replaced by a non-linear function, e.g., sinΔt, tanΔt etc. In the present work, we propose to include one more parameter in the non-linear denominator functions (DFs), e.g., in place of tanΔt, we propose to use ktan(Δt/k), where k∈[c,∞) with c∈R+ which further accelerates the accuracy and reduces the CPU time. We have applied this idea to solve generalized class of non-linear doubly singular BVPs. We observe that there exist some optimal values of k that give much better results. We determine the value of k by experimenting with various DFs on the proposed NSFD schemes. Further it will be interesting to see if we can find out optimal value of k analytically and whether this optimal value is unique. Convergence of the proposed NSFD schemes has also been proved. The results have been verified by several test examples. The results have been compared with other existing methods and matlab bvp4c solver, which shows the applicability and effectiveness of the proposed NSFD schemes.

Unconditionally positive NSFD and classical finite difference schemes for biofilm formation on medical implant using Allen-Cahn equation

Abstract The study of biofilm formation is becoming increasingly important. Microbes that produce biofilms have complicated impact on medical implants. In this paper, we construct an unconditionally positive non-standard finite difference scheme for a mathematical model of biofilm formation on a medical implant. The unknowns in many applications reflect values that cannot be negative, such as chemical component concentrations or population numbers. The model employed here uses the bistable Allen-Cahn partial differential equation, which is a generalization of Fisher’s equation. We study consistency and convergence of the scheme constructed. We compare the performance of our scheme with a classical finite difference scheme using four numerical experiments. The technique used in the construction of unconditionally positive method in this study can be applied to other areas of mathematical biology and sciences. The results here elaborate the benefits of the non-standard approximations over the classical approximations in practical applications.

Positivity and boundedness preserving nonstandard finite difference schemes for solving Volterra’s population growth model

In this work, we extend the Mickens’ methodology to construct nonstandard finite difference (NSFD) schemes preserving positivity and boundedness of the nonlinear Volterra integro-differential population growth model. A rigorously mathematical study for the positivity, boundedness, convergence and error bounds of the proposed NSFD schemes is provided. It is proved that the NSFD schemes not only converge, but also preserve the positivity and boundedness of the integro-differential model for all finite step sizes. Furthermore, the constructed NSFD schemes can be extended to formulate nonstandard discretization schemes for general Volterra integral equations and fractional-order Volterra integro-differential population growth models. Finally, the theoretical results and advantages of the NSFD schemes over standard ones are supported and illustrated by a set of numerical experiments. The numerical experiments show that some typical standard numerical schemes such as the Euler scheme, the second order and classical fourth order Runge–Kutta schemes fail to correctly preserve the positivity and boundedness for some given step sizes. As a result, they can generate numerical approximations that are completely different from the solutions of the integro-differential model. However, these properties are preserved by the NSFD schemes when using the same step sizes.

A Non-Standard Finite Difference Scheme for a Diffusive HIV-1 Infection Model with Immune Response and Intracellular Delay

In this paper, we propose and study a diffusive HIV infection model with infected cells delay, virus mature delay, abstract function incidence rate and a virus diffusion term. By introducing the reproductive numbers for viral infection R0 and for CTL immune response number R1, we show that R0 and R1 act as threshold parameter for the existence and stability of equilibria. If R0≤1, the infection-free equilibrium E0 is globally asymptotically stable, and the viruses are cleared; if R1≤1<R0, the CTL-inactivated equilibrium E1 is globally asymptotically stable, and the infection becomes chronic but without persistent CTL response; if R1>1, the CTL-activated equilibrium E2 is globally asymptotically stable, and the infection is chronic with persistent CTL response. Next, we study the dynamic of the discreted system of our model by using non-standard finite difference scheme. We find that the global stability of the equilibria of the continuous model and the discrete model is not always consistent. That is, if R0≤1, or R1≤1<R0, the global stability of the two kinds model is consistent. However, if R1>1, the global stability of the two kinds model is not consistent. Finally, numerical simulations are carried out to illustrate the theoretical results and show the effects of diffusion factors on the time-delay virus model.

Analysis of the Epidemic Biological Model of Tuberculosis (TB) via Numerical Schemes

Tuberculosis (TB) is caused by bacillus Mycobacterium tuberculosis (MTB). In this study, a mathematical model of tuberculosis (TB) is analyzed. The numerical behaviour of the considered model is analyzed including basic reproduction number and stability. We applied three numerical techniques to this model, i.e., nonstandard finite difference (NSFD) scheme, Runge–Kutta method of order 4(RK-4), and forward Euler (FD) scheme. NSFD scheme preserves all the essential properties of the model. Acquired results corroborate that NSFD scheme converges for each step size. While the other two schemes failed to preserve some properties of the model such as positivity and convergence. A graphical comparison presented in this study confirms the numerical stability of the NSFD technique shown here is maintained over a large area.

Mathematical modelling and nonstandard finite scheme analysis for an Ebola model transmission with information and voluntary isolation

A major Ebola outbreak occurs in West Africa since March 2014, being the deadliest epidemic in history. On August 2014, the World Health Organization (WHO) declared the outbreak public health emergency of international concern. In this work, we propose an epidemic model to study the transmission dynamics of Ebola virus disease (EVD) with high- and low-risk susceptible populations. Our aim is to analyse and assess the impacts of positive and negative information conveyed on the field about EVD spreading. The model considers two different groups of susceptible individuals depending on the level of information about the disease spreading. We provide a theoretical study of the model, compute the disease-free equilibrium (DFE) and derive the basic reproduction number that determines the extinction and the persistence of the disease. Also, we compute equilibria and study their stability. More precisely, we show that the DFE is globally asymptotically stable whenever . Conversely, when is greater than one, the DFE becomes unstable and a unique endemic equilibrium arises which is locally asymptotically stable. These results are obtained through the construction of suitable Lyapunov functions combined with Lyapunov–LaSalle techniques and the search for the eigenvalues of the Jacobian matrix, respectively. We design a nonstandard finite difference scheme, which preserves the essential properties of the continuous system. We perform sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission and prevalence. Furthermore, we provide numerical simulations to support the analytical results.

An Efficient Non-standard Finite Difference Scheme for Solving Distributed Order Time Fractional Reaction–Diffusion Equation

An Efficient Non-standard Finite Difference Scheme for Solving Distributed Order Time Fractional Reaction–Diffusion Equation

Pattern formation in the Holling–Tanner predator–prey model with predator-taxis. A nonstandard finite difference approach

A characteristic feature of living organisms is their response to the environment in search for food or reproduction opportunities. This paper is devoted to the investigation of the pattern formation of the Holling-Tanner predator–prey model with predator-taxis. We first summarise the qualitative properties of the model where a threshold for the appearance of pattern formation is specified. Then we design and analyse a coupled nonstandard finite difference and finite volume scheme for the proposed model. Numerical simulations are provided to support theoretical findings.