Nonstandard Finite Difference Research Articles

Exploring local and global stability of COVID-19 through numerical schemes

Respiratory sensitivity and pneumonia are possible outcomes of the coronavirus (COVID-19). Surface characteristics like temperature and sunshine affect how long the virus survives. This research article analyzes COVID-19 mathematical model behavior based on symptomatic and non-symptomatic individuals. In the reproductive model, the best result indicates the intensity of the epidemic. Our model remained stable at a certain point under controlled conditions after we evaluated a specific element. This approach is in place of traditional approaches such as Euler’s and Runge–Kutta’s. An unusual numerical approach known as the non-standard finite difference (NSFD) scheme is used in this article. This numerical approach gives us positivity. A dependable numerical analysis allowed us to evaluate different approaches and verify our theoretical results. Unlike the widely used Euler and RK4 approaches, we investigated the benefits of implementing NSFD schemes. By numerically simulating COVID-19 in a variety of scenarios, we demonstrated how our theoretical concepts work. The simulation findings support the usefulness of both approaches.

Nonstandard Finite Difference Time Domain Methodology to Simulate Light Propagation in Nonlinear Materials

We extend the nonstandard (NS) finite difference time domain (FDTD) methodology, originally developed to solve Maxwell’s equations in linear materials, to nonlinear ones. We validate it by computing harmonics generation in a nonlinear dielectric and comparing with theory. The methodology also applies to the quantum electrodynamics that describes the interaction of charged particles with electromagnetic fields, and also to the Ginzburg-Landau model of superconductivity.

Nonstandard finite difference schemes for linear and non-linear Fokker–Planck equations

Nonstandard finite difference schemes for linear and non-linear Fokker–Planck equations

An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation

The Huxley equation, which is a nonlinear partial differential equation, is used to describe the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon. This equation, just like many other nonlinear equations, is often very difficult to analyze because of the presence of the nonlinearity term, which is always very difficult to approximate. This paper aims to design a reliable scheme that consists of a combination of the nonstandard finite difference in time method, the Galerkin method and the compactness methods in space variables. This method is used to show that the solution of the problem exists uniquely. The a priori estimate from the existence process is applied to the scheme to show that the numerical solution from the scheme converges optimally in the L2 as well as the H1 norms. We proceed to show that the scheme preserves the decaying properties of the exact solution. Numerical experiments are introduced with a chosen example to validate the proposed theory.

Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations

Abstract We solve two numerical experiments described by 2D nonconstant coefficient advection–diffusion equations with specified initial and boundary conditions. Three finite difference methods, namely Lax–Wendroff, Du-Fort–Frankel and a nonstandard finite difference scheme, are derived and used to solve the two problems, whereby only the first problem has an exact solution. Stability analysis is performed to obtain a range of values of the time step size at a fixed spatial step size. We obtain the rate of convergence in space when the three methods are used to solve Problem 1. Computational times of the three algorithms are computed for Problem 1. Results are displayed for the two problems using the three methods at times T = 1.0 T=1.0 and T = 5.0 T=5.0 . The main novelty is the stability analysis, which is not straightforward as we are working with numerical methods discretising 2D nonconstant coefficient advection–diffusion equation where many parameters are involved. The second highlight is to determine the most efficient scheme from the three methods. Third, there are very few published studies on analysis and use of numerical methods to solve nonconstant coefficient advection–diffusion equations, and this is one of the very few rare articles treating such topics.

High-order nonstandard finite difference methods preserving dynamical properties of one-dimensional dynamical systems

High-order nonstandard finite difference methods preserving dynamical properties of one-dimensional dynamical systems

Nonstandard Finite Difference Scheme for the Epidemic Model with Vaccination

Nonstandard Finite Difference Scheme for the Epidemic Model with Vaccination

Precision in disease dynamics: Finite difference solutions for stochastic epidemics with treatment cure and partial immunity

The complex and ever-changing characteristics of epidemic modelling, particularly when considering random elements, provide a substantial obstacle in creating precise and practical numerical methods for solving differential equations. This study contributes to this effort by introducing an innovative finite difference method for linear and non-linear stochastic and deterministic differential equations. This scheme expands explicitly upon the Euler Maruyama method, improving its precision for the deterministic aspect while ensuring coherence in dealing with stochastic terms. This contribution provides a numerical scheme that can be used to find solutions to linear and non-linear stochastic and deterministic differential equations. The scheme can be considered as the extension of the Euler Maruyama scheme for solving stochastic differential equations. The Euler Maruyama scheme offers a first-order accuracy of the deterministic model. Still, this scheme provides second-order accuracy for the deterministic part, whereas the integration of stochastic terms is the same in both schemes. The scheme is employed in a stochastic diffusive epidemic model with the effect of treatment, cure, and partial immunity. The comparison of the proposed scheme with the existing nonstandard finite difference method is made, and it is shown that the proposed scheme performs better than the nonstandard finite difference method in accuracy for the deterministic differential equations. It is also demonstrated that susceptible people rise whereas infected and recovered people decline by enhancing treatment cure rate. How does the cure rate of the treatment influence the number of the three populations, i.e., S(t), I(t), and R(t)? The results from the numerical simulation have provided useful insights into the dynamics of the epidemic model under various settings. This is particularly useful for influencing any public health plan and intervention. Thus, this work contributes numerical approaches and is an essential tool for epidemiological studies.

A simple model of nutrient recycling and dormancy in a chemostat: Mathematical analysis and a second-order nonstandard finite difference method

A chemostat is an apparatus that sustains a homogeneous environment through continuous inflow and outflow. Presented is a chemostat model that characterizes the dynamics of dormancy-capable microorganisms. This model of coupled systems of nonlinear ordinary differential equations (ODEs) can apply to various types of organisms, such as different species of bacteria, archaea, algae, fungi, viruses, and protozoa. However, these species reside in different environments and rely on different sets of nutrients. Thus, the model adapts to each species’ limiting nutrient through nutrient recycling. This paper includes a complete stability analysis, supporting phase plane portraits, and accompanying bifurcation diagrams. The paper also proposes an advanced second-order, positivity-preserving, and elementary stable nonstandard finite difference method for solving the mathematical model. Series of numerical simulations are presented that support the theoretical results and explore different biological scenarios. The stability analysis reveals that (1) if the overall dilution, death, and conversion are less than the overall growth, both the dormant and active populations persist when introduced to a chemostat; and (2) if the overall dilution, death, and conversion are more than the overall growth, the microorganism population in its entirety will die off when introduced to a chemostat. Furthermore, the model study suggests that neither dormancy nor nutrient recycling provides substantial survival advantages in a basic chemostat when no threat to the active microorganism is present.

Non-Standard Finite Difference and Vieta-Lucas Orthogonal Polynomials for the Multi-Space Fractional-Order Coupled Korteweg-de Vries Equation

This paper focuses on examining numerical solutions for fractional-order models within the context of the coupled multi-space Korteweg-de Vries problem (CMSKDV). Different types of kernels, including Liouville-Caputo fractional derivative, as well as Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, are utilized in the examination. For this purpose, the nonstandard finite difference method and spectral collocation method with the properties of the Shifted Vieta-Lucas orthogonal polynomials are employed for converting these models into a system of algebraic equations. The Newton-Raphson technique is then applied to solve these algebraic equations. Since there is no exact solution for non-integer order, we use the absolute two-step error to verify the accuracy of the proposed numerical results.

A Third-order Two Stage Numerical Scheme and Neural Network Simulations for SEIR Epidemic Model: A Numerical Study

This study focuses on the cutting-edge field of epidemic modeling, providing a comprehensive investigation of a third-order two-stage numerical approach combined with neural network simulations for the SEIR (Susceptible-Exposed-Infectious-Removed) epidemic model. An explicit numerical scheme is proposed in this work for dealing with both linear and nonlinear boundary value problems. The scheme is built on two grid points, or two time levels, and is third-order. The main advantage of the scheme is its order of accuracy in two stages. Third-order precision is not only not provided by most existing explicit numerical approaches in two phases, but it also necessitates the computation of an additional derivative of the dependent variable. The proposed scheme's consistency and stability are also examined and presented. Nonlinear SEIR (susceptible-exposed-infected-recovered) models are used to implement the scheme. The scheme is compared with the non-standard finite difference and forward Euler methods that are already in use. The graph shows that the plan is more accurate than non-standard finite difference and forward Euler methods that are already in use. The solution obtained is then looked at through the lens of the neural network. The neural network is trained using an optimization approach known as the Levenberg-Marquardt backpropagation (LMB) algorithm. The mean square error across the total number of iterations, error histograms, and regression plots are the various graphs that can be created from this process. This work conducts thorough evaluations to not only identify the strengths and weaknesses of the suggested approach but also to examine its implications for public health intervention. The results of this study make a valuable contribution to the continuously developing field of epidemic modeling. They emphasize the importance of employing modern numerical techniques and machine learning algorithms to enhance our capacity to predict and effectively control infectious diseases. Doi: 10.28991/ESJ-2024-08-01-023 Full Text: PDF

Developing computationally efficient optimal control strategies to eradicate Rubella disease

The threat of Rubella virus disease looms large, posing significant risks to public health and emphasizing the urgent need for comprehensive prevention, control, and awareness strategies. We conducted an extensive analysis of a newly developed SEITR deterministic model for the lethal Rubella virus disease. The main objective of our study is to gain deep insights into the disease dynamics and devise an optimal control strategy for the model, utilizing vaccination and treatment as preventive measures. We employed various mathematical techniques to establish the positivity and bounded nature of solutions. The value of threshold parameter is computed using the next-generation method to anticipate future dynamical behavior of the epidemic. The local and global stability of the equilibrium points was successfully assessed. Additionally, we utilized the well-known Non-Standard Finite Difference (NSFD) method to obtain numerical solutions for the Rubella model. A numerical analysis is carried out to assess the efficacy of a constant treatment strategy, and the results are presented through graphical illustrations. The developed model is subjected to sensitivity analysis and the most sensitive parameters are identified. In addition, the bifurcation nature of the model is examined. Subsequently, an optimal control problem is introduced for the model, aiming to determine the best time-dependent strategies for treatment and vaccination. The main goal is to reduce the number of individuals infected within the human population and the cost of controls. Designed optimal control problem and its corresponding optimality conditions of Pontryagin type have been derived. An important aspect of this study is the utilization of the NSFD method, implemented backward in time, to solve the optimal control problem, as opposed to other conventional methods. Numerical simulations were carried out to assess the impact of the applied controls on the dynamics of all classes, both before and after optimization.

Numerical analysis of singularly perturbed parabolic reaction diffusion differential difference equations

The authors present numerical analysis of singularly perturbed parabolic problems. The previous papers in this direction focussed on the convection-diffusion problems with regular type of boundary layers. It is very difficult and almost impossible as stated in the literature, e.g, in Chapter 14 of ‘Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions’ (1996) by Miller, John JH and O'Riordan, Eugene and Shishkin, Grigorii I, to capture singularities due to parabolic layers and develop a robust scheme for reaction-diffusion problems with general values of space shifts. The work is in progress now and this is the first paper in this direction. In this work, we are investigating such problems and develop a robust numerical scheme using Mickens's non-standard finite difference scheme and special mesh. Furthermore, numerical examples have been presented to verify the theoretical findings.

A second-order nonstandard finite difference method for a general Rosenzweig–MacArthur predator–prey model

In this paper, we consider a general Rosenzweig–MacArthur predator–prey model with logistic intrinsic growth of the prey population. We develop the Mickens’ method to construct a dynamically consistent second-order nonstandard finite difference (NSFD) scheme for the general Rosenzweig–MacArthur predator–prey model. The second-order NSFD method is based on a novel nonlocal approximation using right-hand side function weights and nonstandard denominator functions.Through rigorous mathematical analysis, we show that the NSFD method not only preserves two important and prominent dynamical properties of the continuous-time model, namely positivity and asymptotic stability independent of the values of the step size, but also is convergent of order 2. Therefore, it provides a solution to the contradiction between the dynamic consistency and high-order accuracy of NSFD methods.The proposed NSFD method improves positive and elementary stable nonstandard numerical schemes constructed in a previous work of Dimitrov and Kojouharov (2006). Moreover, the present approach can be extended to construct second-order NSFD methods for some classes of nonlinear dynamical systems.Finally, the theoretical insights and advantages of the constructed NSFD scheme are supported by some illustrative numerical simulations.

A generalized model for the population dynamics of a two stage species with recruitment and capture using a nonstandard finite difference scheme

The aim of this work is to formulate and analyze a new and generalized discrete-time population dynamics model for a two-stage species with recruitment and capture factors. This model is derived from a well-known continuous-time population dynamics model of a two-stage species with recruitment and capture developed by Ladino and Valverde and the nonstandard finite difference (NSFD) methodology proposed by Mickens. We establish positivity and asymptotic stability of the proposed discrete-time population dynamics model. As an important consequence, the population dynamics of the new discrete-time model is determined fully. Also, a set of numerical examples is conducted to illustrate the theoretical results and to demonstrate advantages of the new model. The theoretical results and numerical examples show that the proposed discrete-time model not only preserves correctly the population dynamics of the continuous one but is also easy to be implemented. However, some discrete-time models based on the standard Runge–Kutta methods fail to preserve the population dynamics of the continuous-time model. As a result, they generate numerical approximations which are not only non-negative but also unstable.

Nonstandard finite difference method for time-fractional singularly perturbed convection–diffusion problems with a delay in time

In this work, nonstandard finite difference method is presented for the numerical solution of time-fractional singularly perturbed convection–diffusion problems with a delay in time. The time-fractional derivative is considered in the Caputo sense and discretized using Crank–Nicholson technique. Then, a nonstandard finite difference scheme is constructed on a uniform mesh discretization along the spatial direction. The parameter-uniform convergence of the proposed method is proved rigorously and shown to be ɛ-uniform convergent with order of convergence O((Δt)2) along the temporal domain and M−1 along the spatial domain. Finally, the proposed scheme is validated using model examples and the computational results are in agreement with the theoretical expectation.

Extending nonstandard finite difference scheme rules to systems of nonlinear ODEs with constant coefficients

In this paper, we present a reformulation of Mickens' rules for nonstandard finite difference (NSFD) schemes to adapt them to systems of ODEs. We want to compare several methods within the class of NSFD schemes. These methods are expressed as exponential integrators which treat exactly the linear part of the equations. While the classical methods address each equation of a system separately, we propose here to address them as a whole, which involves a reformulation of Mickens' rules. This leads to enhance the results compared to the classical case, even if approximations of the matrix exponentials are used. Precisely, using Cayley–Hamilton theorem we obtain two correction factors, one of which is linked to the size of the system and the other to the nonlinear term, we are thus able to use test cases which show the impact of one or another corrector. We prove the consistency and the zero-stability of the method in the nonlinear case to conclude to the convergence of the method.

A nonstandard compact finite difference method for a truncated Bratu–Picard model

<p>In this paper, we used the nonstandard compact finite difference method to numerically solve one-dimensional truncated Bratu-Picard equations and discussed the convergence analysis of the proposed method. Depending on the parameters in the mentioned equation, it may have no solution, one solution, or two solutions; also, it may have infinitely many solutions. The numerical results show that our method covers all mentioned aspects depending on the parameters in the equation.</p>

Optimization and control in rubella transmission dynamics: A boundedness-preserving numerical model with vaccination

Rubella is an infectious disease that can spread globally. It spreads in the subtropics as well as the tropics. Although it is commonly thought to be a non-fatal condition, there are several circumstances in which it can be fatal. Pregnant women infected with the Rubella virus face a high risk of fetal development. The main goal of this study is to look into a model for the spread of Rubella while considering a vaccination campaign as a control measure. The positivity and boundedness of the re-infection rubella transmission and vaccination model are investigated. The local and global stability of the equilibrium points is examined. The sensitivity of parameters is also investigated. Three different techniques forward Euler, RK-4, and non-standard finite difference (NSFD) method are developed for the numerical solution, and their simulation results are examined. Among these three, the NSFD method is superior due to its convergence and positive behavior for all step size values. For some values of the step size, the other two techniques failed to produce positivity and convergent solutions. Analytically, the proposed model's convergence, positivity, boundedness, and consistency are investigated. Finally, the impact of the Rubella vaccine on infected populations has been examined, revealing that vaccination is one of the most effective ways to prevent rubella transmission.

Nonstandard Computational and Bifurcation Analysis of the Rabies Epidemic Model

This paper aims to investigate some new dynamics of a new model describing the rabies virus dynamics, taking into account the effect of proper vaccination. The model’s population is divided into three main compartments, namely, susceptible S(t), infected I(t), and recovered R(t) individuals. The model is formulated and then the equilibrium points of the model are found. The local and global stabilities of equilibrium points of the proposed model are investigated where conditions of stability are attained in terms of key parameters in the model. Bifurcation analysis is performed for the possible occurrence of codimension‐one bifurcations in the model. In addition, bifurcation surfaces are plotted in the space of parameters in the model. For the numerical verification, a nonstandard finite difference method is adapted for solving the model and the accurate results of numerical simulations are depicted to reveal the dynamics of the model. The method provides realistic data for the model and these data can be used to predict the spread of the virus and to provide insight into proper procedures and control measures that can be taken.