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Comprehensive Numerical Analysis of Time-Fractional Reaction–Diffusion Models with Applications to Chemical and Biological Phenomena

Author

Listed:
  • Kolade M. Owolabi

    (Department of Mathematical Sciences, Federal University of Technology, Akure PMB 704, Ondo State, Nigeria)

  • Sonal Jain

    (School of Technology, Woxsen University, Hyderabad 502345, Telangana, India)

  • Edson Pindza

    (Department of Decision Sciences, College of Economic and Management Sciences, University of South Africa (UNISA), Pretoria 0003, South Africa)

  • Eben Mare

    (Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa)

Abstract

This paper aims to present a robust computational technique utilizing finite difference schemes for accurately solving time fractional reaction–diffusion models, which are prevalent in chemical and biological phenomena. The time-fractional derivative is treated in the Caputo sense, addressing both linear and nonlinear scenarios. The proposed schemes were rigorously evaluated for stability and convergence. Additionally, the effectiveness of the developed schemes was validated through various linear and nonlinear models, including the Allen–Cahn equation, the KPP–Fisher equation, and the Complex Ginzburg–Landau oscillatory problem. These models were tested in one-, two-, and three-dimensional spaces to investigate the diverse patterns and dynamics that emerge. Comprehensive numerical results were provided, showcasing different cases of the fractional order parameter, highlighting the schemes’ versatility and reliability in capturing complex behaviors in fractional reaction–diffusion dynamics.

Suggested Citation

  • Kolade M. Owolabi & Sonal Jain & Edson Pindza & Eben Mare, 2024. "Comprehensive Numerical Analysis of Time-Fractional Reaction–Diffusion Models with Applications to Chemical and Biological Phenomena," Mathematics, MDPI, vol. 12(20), pages 1-26, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:20:p:3251-:d:1500717
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    References listed on IDEAS

    as
    1. Khater, Mostafa M.A. & Mohamed, Mohamed S. & Attia, Raghda A.M., 2021. "On semi analytical and numerical simulations for a mathematical biological model; the time-fractional nonlinear Kolmogorov–Petrovskii–Piskunov (KPP) equation," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    2. Alqhtani, Manal & Owolabi, Kolade M. & Saad, Khaled M., 2022. "Spatiotemporal (target) patterns in sub-diffusive predator-prey system with the Caputo operator," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    3. Rocco, Andrea & West, Bruce J., 1999. "Fractional calculus and the evolution of fractal phenomena," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 265(3), pages 535-546.
    4. Rahimabadi, Arsalan & Benali, Habib, 2023. "Extended fractional-polynomial generalizations of diffusion and Fisher–KPP equations on directed networks," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    5. Owolabi, Kolade M. & Jain, Sonal, 2023. "Spatial patterns through diffusion-driven instability in modified predator–prey models with chaotic behaviors," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
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