Contributions to the mathematics of the nonstandard finite difference method and applications

Anguelov, Roumen; Lubuma, Jean M.‐S.

doi:10.1002/num.1025

Cited by 210 publications

(201 citation statements)

References 14 publications

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“…Examples to that effect for similar systems can be found in [12]. A general discussion of the issue can be found in [7,46,48]. In Fig.…”

Section: Numerical Simulationsmentioning

confidence: 84%

“…According to the approach of [44], formalized in [7], the two numerical methods (24) and (28) are non-standard finite difference schemes for the following reasons: (a) The standard denominator h of the discrete derivatives is replaced by the more complex function φ(h), which satisfies the requirement (23). (This denominator function is expected to capture the essential features of the dynamical system); (b) Nonlinear terms are approximated in a nonlocal way by using more than one point of the mesh.…”

Section: Remarkmentioning

confidence: 99%

“…The second direction in the validation of the discrete models is to avoid spurious or ghost solutions in the sense of the next definition [7], which describes the minimum qualitative property that a reliable scheme should have.…”

Section: Remarkmentioning

confidence: 99%

“…To that end, we use the Nonstandard Finite Difference (NSFD) method, which was initiated more than two decades ago by R.E. Mickens and has, since the publication of his monograph [44], shown great potential in replicating the dynamics of systems that arise in a variety of areas in science and engineering (see, e.g., the edited volumes [46,47], the papers [4,7,25,26], and also [31] for an overview and applications in Biosciences). Furthermore, the usefulness and reliability of the NSFD method in epidemiology is attested by the ever growing number of works over the past few years (see, e.g., [21,22,27,28,53]).…”

Section: Introductionmentioning

confidence: 99%

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Dynamically consistent nonstandard finite difference schemes for epidemiological models

Anguelov

Dumont

Lubuma

et al. 2014

Journal of Computational and Applied Mathematics

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This work is the numerical analysis and computational companion of the paper by Kamgang and Sallet (Math. Biosc. 213 (2008), pp. 1-12) where threshold conditions for epidemiological models and the global stability of the disease-free equilibrium (DFE) are studied. We establish a discrete counterpart of the main continuous result that guarantees the global asymptotic stability (GAS) of the DFE for general epidemiological models. Then, we design nonstandard finite difference (NSFD) schemes in which the Metzler matrix structure of the continuous model is carefully incorporated and both Mickens' rules (World Scientific, Singapore, 1994) on the denominator of the discrete derivative and the nonlocal approximation of nonlinear terms are used in an innovative way. As a result of these strategies, our NSFD schemes are proved to be dynamically consistent with the continuous model, i.e., they replicate their basic features, including the GAS of the DFE, the linear stability of the endemic equilibrium (EE), the positivity of the solutions, the dissipativity of the system, and its inherent conservation law. The general analysis is made detailed for the MSEIR model for which the NSFD theta method is implemented, with emphasis on the computational aspects such as its convergence, or local truncation error. Numerical simulations that illustrate the theory are provided.

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“…Examples to that effect for similar systems can be found in [12]. A general discussion of the issue can be found in [7,46,48]. In Fig.…”

Section: Numerical Simulationsmentioning

confidence: 84%

Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dynamically consistent nonstandard finite difference schemes for epidemiological models

Anguelov

Dumont

Lubuma

et al. 2014

Journal of Computational and Applied Mathematics

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“…Here, we design a nonstandard finite difference (NSFD) scheme to approximate the solution of the continuous model. The scheme is constructed based on two Mickens' rules [15] and as singled out in [2]. To the best of our knowledge, this is the first time to present the rabies model between human and dog populations with its dynamically consistent numerical approximation.…”

Section: Introductionmentioning

confidence: 99%

Analysis and dynamically consistent nonstandard discretization for a rabies model in humans and dogs

Chapwanya

Lubuma

Terefe

2015

RACSAM

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Rabies is a fatal disease in dogs as well as in humans. A possible model to represent rabies transmission dynamics in human and dog populations is presented. The next generation matrix operator is used to determine the threshold parameter R 0 , that is the average number of new infective individuals produced by one infective individual introduced into a completely susceptible population. If R 0 < 1, the disease-free equilibrium is globally asymptotically stable, while it is unstable and there exists a locally asymptotically stable endemic equilibrium when R 0 > 1. A nonstandard finite difference scheme that replicates the dynamics of the continuous model is proposed. Numerical tests to support the theoretical analysis are provided.

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

Díaz‐Rodríguez¹,

González-Parra²,

Arenas³

2009

Int. J. Numer. Meth. Biomed. Engng.

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SUMMARYIn 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.

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Contributions to the mathematics of the nonstandard finite difference method and applications

Cited by 210 publications

References 14 publications

Dynamically consistent nonstandard finite difference schemes for epidemiological models

Dynamically consistent nonstandard finite difference schemes for epidemiological models

Analysis and dynamically consistent nonstandard discretization for a rabies model in humans and dogs

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

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