Extended fractional-polynomial generalizations of diffusion and Fisher–KPP equations on directed networks

Rahimabadi, Arsalan; Benali, Habib

doi:10.1016/j.chaos.2023.113771

Chaos, Solitons & Fractals

2023

DOI: 10.1016/j.chaos.2023.113771

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Extended fractional-polynomial generalizations of diffusion and Fisher–KPP equations on directed networks

Arsalan Rahimabadi

Habib Benali

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2024

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Cited by 3 publications

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

Owolabi,

Jain,

Pindza

et al. 2024

Mathematics

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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.

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

Owolabi,

Jain,

Pindza

et al. 2024

Mathematics

View full text Add to dashboard Cite

show abstract

Nonlinear diffusion on networks: Perturbations and consensus dynamics

Bonetto,

Kojakhmetov

2024

NHM

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<p>In this paper, we study a class of equations representing nonlinear diffusion on networks. A particular instance of our model could be seen as a network equivalent of the porous-medium equation. We are interested in studying perturbations of such a system and describing the consensus dynamics. The nonlinearity of the equations gives rise to potentially intricate structures of equilibria that could intersect the consensus space, creating singularities. For the unperturbed case, we characterize the sets of equilibria by exploiting the symmetries under group transformations of the nonlinear vector field. Under small perturbations, we obtain a slow-fast system. Thus, we analyze the slow-fast dynamics near the singularities on the consensus space. The analysis at this stage is carried out for complete networks, allowing a detailed characterization of the system. We provide a linear approximation of the intersecting branches of equilibria at the singular points; as a consequence, we show that, generically, the singularities on the consensus space turn out to be <italic>transcritical</italic>. We prove under local assumptions the existence of canard solutions. For generic graph structures, assuming more strict conditions on the perturbation, we prove the existence of a maximal canard, which coincides with the consensus subspace. In addition, we validate by numerical simulations the principal findings of our main theory, extending the study to non-complete graphs. Moreover, we show how the delayed loss of stability associated with the canards induces transient spatio-temporal patterns.</p>

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Nonlinear diffusion on networks: Perturbations and consensus dynamics

Bonetto,

Kojakhmetov

2024

NHM

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Extended fractional-polynomial generalizations of diffusion and Fisher–KPP equations on directed networks

Cited by 3 publications

References 49 publications

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

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

Nonlinear diffusion on networks: Perturbations and consensus dynamics

Nonlinear diffusion on networks: Perturbations and consensus dynamics

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