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On the nonstandard finite difference method for reaction–diffusion models

Author

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  • Pasha, Syed Ahmed
  • Nawaz, Yasir
  • Arif, Muhammad Shoaib

Abstract

The nonstandard finite difference (NSFD) method is an elegant approach in that it overcomes the numerical instability and bias exhibited by standard finite difference methods for numerically solving nonlinear differential equations. In addition, the NSFD method preserves some qualitative features of the continuous-time model such as boundedness and positivity. But for an important class of models which include diffusion and reaction–diffusion systems that appear in a number of application domains including epidemiology, ecology, and finance, recently introduced NSFD schemes do not guarantee first-order temporal accuracy or consistency. In this paper, we first show this for a reaction–diffusion epidemic model. We then propose an alternative NSFD scheme that guarantees first-order accuracy in time and second-order accuracy in space whilst preserving positivity of the solution. Stability and consistency analyses of the proposed scheme are then presented. To demonstrate the performance of the proposed scheme we show a comparison with the existing NSFD approach for three examples which confirm the superiority of our proposed approach.

Suggested Citation

  • Pasha, Syed Ahmed & Nawaz, Yasir & Arif, Muhammad Shoaib, 2023. "On the nonstandard finite difference method for reaction–diffusion models," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    • Handle: RePEc:eee:chsofr:v:166:y:2023:i:c:s0960077922011080
    DOI: 10.1016/j.chaos.2022.112929
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    References listed on IDEAS

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    1. Elaiw, A.M. & Alshaikh, M.A., 2020. "Stability preserving NSFD scheme for a general virus dynamics model with antibody and cell-mediated responses," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    2. Company, Rafael & Jódar, Lucas & Pintos, José-Ramón, 2012. "A consistent stable numerical scheme for a nonlinear option pricing model in illiquid markets," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(10), pages 1972-1985.
    3. 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.
    4. Dimitrov, Dobromir T. & Kojouharov, Hristo V., 2008. "Nonstandard finite-difference methods for predator–prey models with general functional response," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 78(1), pages 1-11.
    5. Mickens, Ronald E., 2016. "NSFD scheme for acoustic propagation with the linearized Euler equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 127(C), pages 189-193.
    6. Jang, Junyoung & Kwon, Hee-Dae & Lee, Jeehyun, 2020. "Optimal control problem of an SIR reaction–diffusion model with inequality constraints," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 171(C), pages 136-151.
    7. Ahmed, Nauman & Rafiq, Muhammad & Adel, Waleed & Rezazadeh, Hadi & Khan, Ilyas & Nisar, Kottakkaran Sooppy, 2020. "Structure Preserving Numerical Analysis of HIV and CD4+T-Cells Reaction Diffusion Model in Two Space Dimensions," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    8. Wood, Daniel T. & Kojouharov, Hristo V. & Dimitrov, Dobromir T., 2017. "Universal approaches to approximate biological systems with nonstandard finite difference methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 133(C), pages 337-350.
    9. Hongpeng Guo & Zhiming Guo, 2021. "Traveling Waves Solutions for Delayed Temporally Discrete Non-Local Reaction-Diffusion Equation," Mathematics, MDPI, vol. 9(16), pages 1-20, August.
    10. Liu, Hong & Yong, Jiongmin, 2005. "Option pricing with an illiquid underlying asset market," Journal of Economic Dynamics and Control, Elsevier, vol. 29(12), pages 2125-2156, December.
    11. Ahmed, Nauman & Ali, Mubasher & Baleanu, Dumitru & Rafiq, Muhammad & Rehman, Muhammad Aziz ur, 2020. "Numerical analysis of diffusive susceptible-infected-recovered epidemic model in three space dimension," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    12. 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.
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