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Nonstandard finite difference method by nonlocal approximation

Author

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  • Anguelov, Roumen
  • Lubuma, Jean M.-S.

Abstract

Two types of monotonic properties of solutions of differential equations are discussed and general finite difference schemes, which are stable with respect to these properties are investigated. Apart from being elementary stable, these schemes are also shown to preserve qualitative properties of nonhyperbolic fixed points of the differential equations. From the practical point of view, a systematic procedure based on nonlocal approximation, is proposed for the construction of qualitatively stable nonstandard finite difference schemes for the logistic equation, the combustion model and the reaction-diffusion equation.

Suggested Citation

  • Anguelov, Roumen & Lubuma, Jean M.-S., 2003. "Nonstandard finite difference method by nonlocal approximation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 61(3), pages 465-475.
    • Handle: RePEc:eee:matcom:v:61:y:2003:i:3:p:465-475
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    Cited by:

    1. Nasser Hassan Sweilam & Seham Mahyoub Al-Mekhlafi & Taghreed Abdul Rahman Assiri, 2017. "Numerical Study for Time Delay Multistrain Tuberculosis Model of Fractional Order," Complexity, Hindawi, vol. 2017, pages 1-14, July.
    2. Jódar, Lucas & Villanueva, Rafael J. & Arenas, Abraham J. & González, Gilberto C., 2008. "Nonstandard numerical methods for a mathematical model for influenza disease," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 622-633.
    3. Pius W. M. Chin & Claude R. B. Moutsinga & Khadijo R. Adem, 2024. "An Analysis of the Nonstandard Finite Difference and Galerkin Methods Applied to the Huxley Equation," Mathematics, MDPI, vol. 12(6), pages 1-18, March.
    4. 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.
    5. Vasily E. Tarasov, 2024. "Exact Finite-Difference Calculus: Beyond Set of Entire Functions," Mathematics, MDPI, vol. 12(7), pages 1-37, March.
    6. Zhang, Hong & Georgescu, Paul & Hassan, Adamu Shitu, 2016. "Mathematical insights and integrated strategies for the control of Aedes aegypti mosquito," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 1059-1089.
    7. Rihan, F.A. & Velmurugan, G., 2020. "Dynamics of fractional-order delay differential model for tumor-immune system," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    8. J. Sunday & Y. Skwame & T. Y. Kyagya, 2017. "Simulation of Riccati Differential Equations by Nonlocal Approximation of Nonlinear Terms and Reconstruction of Denominator Functions," Academic Journal of Applied Mathematical Sciences, Academic Research Publishing Group, vol. 3(7), pages 62-68, 07-2017.

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