A second-order modified nonstandard theta method for one-dimensional autonomous differential equations

Kojouharov, Hristo V.; Roy, Souvik; Gupta, Madhu; Alalhareth, Fawaz K.; Slezak, John M.

doi:10.1016/j.aml.2020.106775

Cited by 27 publications

(16 citation statements)

References 7 publications

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“…It is easy to observe that the standard Euler scheme generated the numerical approximation oscillates around the equilibrium position, the approximation generated by the RK2 scheme converges to a spurious equilibrium point; however, the NSFD scheme preserves the dynamics of the logistic equation. This result is completely consistent with recognized results on NSFD schemes for differential equations [2,11,14,15,16,16,17,18,20,21,23]. On the other hand, it is easy to observe that the NSFD scheme (4) is more accurate than the PESNSFD method.…”

Section: The Logistic Differential Equationsupporting

confidence: 89%

“…The method is based on novel non-local approximations for right-hand side functions of differential equations in combination with nonstandard denominator functions. The obtained results not only resolve the contradiction between the dynamic consistency and high-order accuracy of NSFD schemes but also improve and extend the well-known results constructed in [6,11,23].…”

Section: Introductionsupporting

confidence: 77%

“…However, most of NSFD schemes are only of first order accuracy. For this reason, there have been many attempts to formulate higher-order NSFD schemes for ordinary differential equations (ODEs) [4,6,7,11,12,13].…”

Section: Introductionmentioning

confidence: 99%

“…In very recent works [6,11], second-order NSFD methods preserving the local asymptotic stability (LAS) of one-dimensional autonomous ODEs have been introduced. These results partially resolved the contradiction between the dynamic consistency and high-order accuracy of NSFD schemes; however, the constructed methods did not preserve the positivity of solutions of differential equations.…”

Section: Introductionmentioning

confidence: 99%

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A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems

Hoang

2021

Preprint

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In this work, a novel second-order nonstandard finite difference (NSFD) method that preserves simultaneously the positivity and local asymptotic stability of one-dimensional autonomous dynamical systems is introduced and analyzed. This method is based on novel non-local approximations for right-hand side functions of differential equations in combination with nonstandard denominator functions. The obtained results not only resolve the contradiction between the dynamic consistency and high-order accuracy of NSFD methods but also improve and extend some well-known results that have been published recently in [Applied Mathematics Letters 112(2021) 106775], [AIP Conference Proceedings 2302(2020) 110003] and [Applied Mathematics Letters 50(2015) 78-82]. Furthermore, as a simple but important application, we apply the constructed NSFD method for solving the logistic, sine, cubic, and Monod equations; consequently, the NSFD schemes constructed in the earlier work [Journal of Computational and Applied Mathematics 110(1999) 181-185] are improved significantly. Finally, we report some numerical experiments to support and illustrate the theoretical assertions as well as advantages of the constructed NSFD method.

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Section: The Logistic Differential Equationsupporting

confidence: 89%

Section: Introductionsupporting

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems

Hoang

2021

Preprint

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“…Hence, NSFD schemes are effective to simulate dynamics of dynamical differential models over long time periods. Because of this, nowadays NSFD schemes have been recognized as one of the effective approaches to solve differential equation equations arising in theory and practice [1,5,10,13,23,25,26,39,53,56,57,62,63]. In recent works [15,16,17,18,19,20,31,32], we have successfully developed the Mickens's methodology to construct NSFD schemes for some mathematical models arising in biology and epidemiology.…”

Section: Introductionmentioning

confidence: 99%

Dynamical analysis of a generalized hepatitis B epidemic model and its dynamically consistent discrete model

Hoang¹

2021

Preprint

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The aim of this work is to study qualitative dynamical properties of a generalized hepatitis B epidemic model and its dynamically consistent discrete model. Positivity, boundedness, the basic reproduction number and asymptotic stability properties of the model are analyzed rigorously. By the Lyapunov stability theory and the Poincare-Bendixson theorem in combination with the Bendixson-Dulac criterion, we show that a disease-free equilibrium point is globally asymptotically stable if the basic reproduction number R 0 ≤ 1 and a diseaseendemic equilibrium point is globally asymptotically stable whenever R 0 > 1. Next, we apply the Mickens' methodology to propose a dynamically consistent nonstandard finite difference (NSFD) scheme for the continuous model. By rigorously mathematical analyses, it is proved that the constructed NSFD scheme preserves essential mathematical features of the continuous model for all finite step sizes. Finally, numerical experiments are conducted to illustrate the theoretical findings and to demonstrate advantages of the NSFD scheme over standard ones. The obtained results in this work not only improve but also generalize some existing recognized works.

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A Higher-Order NSFD Method for a Simple Growth Model in the Chemostat

Alalhareth

Kojouharov

2022

Mathematics in Industry

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A second-order modified nonstandard theta method for one-dimensional autonomous differential equations

Cited by 27 publications

References 7 publications

A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems

A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems

Dynamical analysis of a generalized hepatitis B epidemic model and its dynamically consistent discrete model

A Higher-Order NSFD Method for a Simple Growth Model in the Chemostat

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