A new two-dimensional macroscopic local fractional viscous diffusive model of vehicular traffic flow

Bhawna Pokhriyal; Pranay Goswami; Kranti Kumar; Abdalla S. Mahmoud; Mohammed Abdalbagi; Saad Althobaiti; Bhawna Pokhriyal; Pranay Goswami; Kranti Kumar; Abdalla S. Mahmoud; Mohammed Abdalbagi; Saad Althobaiti

doi:10.3934/math.2025928

AIMS Mathematics

2025, Volume 10, Issue 9: 20782-20804. doi: 10.3934/math.2025928

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Research article

A new two-dimensional macroscopic local fractional viscous diffusive model of vehicular traffic flow

1.
Dr. B. R. Ambedkar University Delhi, Delhi 110006, India
2.
Department of Science and Technology, University College-Ranyah, Taif University, Taif 21944, Saudi Arabia

Received: 29 March 2025 Revised: 27 August 2025 Accepted: 02 September 2025 Published: 09 September 2025
MSC : 26A27, 26A33, 35C10, 35R11, 74G10

In this paper, we proposed a two-dimensional macroscopic local fractional viscous-diffusive model of vehicular traffic flow, derived from the two-dimensional fractal Navier-Stokes equation. The model was formulated by combining the derived momentum equation with the fractal LWR framework. The proposed fractal dynamic velocity equation incorporates convection, anticipation, relaxation, diffusion, and viscosity as governing parameters. A stability analysis was performed, and an analytical solution to the model was obtained. Certain aspects of multilane traffic flow are further explored through illustrative examples. The solutions are presented and discussed using graphical representations, which demonstrate how non-differentiable traffic density and speed functions evolve dynamically. The study highlights the influence of viscous-diffusive effects on traffic flow, showing that road narrowing increases viscosity and diffusion, thereby reducing traffic speed, whereas these factors have a comparatively smaller impact on wider roads.
- local fractional calculus,
- macroscopic model,
- two-dimensional LWR model,
- fractal Laplace transform,
- traffic flow model
Citation: Bhawna Pokhriyal, Pranay Goswami, Kranti Kumar, Abdalla S. Mahmoud, Mohammed Abdalbagi, Saad Althobaiti. A new two-dimensional macroscopic local fractional viscous diffusive model of vehicular traffic flow[J]. AIMS Mathematics, 2025, 10(9): 20782-20804. doi: 10.3934/math.2025928

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Abstract

In this paper, we proposed a two-dimensional macroscopic local fractional viscous-diffusive model of vehicular traffic flow, derived from the two-dimensional fractal Navier-Stokes equation. The model was formulated by combining the derived momentum equation with the fractal LWR framework. The proposed fractal dynamic velocity equation incorporates convection, anticipation, relaxation, diffusion, and viscosity as governing parameters. A stability analysis was performed, and an analytical solution to the model was obtained. Certain aspects of multilane traffic flow are further explored through illustrative examples. The solutions are presented and discussed using graphical representations, which demonstrate how non-differentiable traffic density and speed functions evolve dynamically. The study highlights the influence of viscous-diffusive effects on traffic flow, showing that road narrowing increases viscosity and diffusion, thereby reducing traffic speed, whereas these factors have a comparatively smaller impact on wider roads.

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