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

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  • Hoang, Manh Tuan

Abstract

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.

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  • Hoang, Manh Tuan, 2022. "Positivity and boundedness preserving nonstandard finite difference schemes for solving Volterra’s population growth model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 359-373.
    • Handle: RePEc:eee:matcom:v:199:y:2022:i:c:p:359-373
    DOI: 10.1016/j.matcom.2022.04.003
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    References listed on IDEAS

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    1. Tuan Hoang, Manh & Nagy, A.M., 2019. "Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite difference schemes," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 24-34.
    2. Mickens, Ronald E., 2003. "A nonstandard finite difference scheme for the diffusionless Burgers equation with logistic reaction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 62(1), pages 117-124.
    3. Hoang, Manh Tuan, 2022. "Reliable approximations for a hepatitis B virus model by nonstandard numerical schemes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 32-56.
    4. Arenas, Abraham J. & González-Parra, Gilberto & Chen-Charpentier, Benito M., 2016. "Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 121(C), pages 48-63.
    5. Aderogba, A.A. & Fabelurin, O.O. & Akindeinde, S.O. & Adewumi, A.O. & Ogundare, B.S., 2020. "Nonstandard finite difference approximation for a generalized Fins problem," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 183-191.
    6. Adamu, Elias M. & Patidar, Kailash C. & Ramanantoanina, Andriamihaja, 2021. "An unconditionally stable nonstandard finite difference method to solve a mathematical model describing Visceral Leishmaniasis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 171-190.
    7. Mickens, Ronald E., 2005. "A nonstandard finite difference scheme for a PDE modeling combustion with nonlinear advection and diffusion," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 69(5), pages 439-446.
    8. Khalsaraei, Mohammad Mehdizadeh & Shokri, Ali & Ramos, Higinio & Heydari, Shahin, 2021. "A positive and elementary stable nonstandard explicit scheme for a mathematical model of the influenza disease," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 397-410.
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