A FRACTIONAL NONLOCAL PARTIAL DIFFERENTIAL EQUATION FRAMEWORK FOR ADAPTIVE TRAFFIC FLOW DYNAMICS

Abstract

Many traffic flow-based nonlocal models consist of microscopic Ordinary Differential Equations (ODEs) and macroscopic Partial Differential Equations (PDEs). In this paper, a novel mathematical model based on fractional and nonlocal partial differential equations (PDEs) is proposed to capture adaptive traffic flow dynamics in complex urban networks. Conventional models like the Lighthill–Whitham–Richards (LWR) model fail to capture interactions and memory impacts, relying on systems-based real traffic. To address these restrictions, a system-based fractional-order nonlocal PDE integrating the interactions-based spatial distribution and diffusion coefficients-based time-dependent adaptivity is proposed. The proposed model combines fractional derivatives for describing inherited traffic environmental conditions and nonlocal CORS for representing driver anticipation influences. Analytical properties consisting of steady state, presence, and solutions-based boundedness are considered over appropriate assumptions. A finite difference discretization of the numerical approach and an approximation method-based Grünwald–Letnikov scheme are modeled for simulation. MATLAB-based experiments illustrate that the proposed model captures a sharp change in pressure in a narrow region traveling through a medium, especially air, caused by an explosion or by a body moving faster than sound propagation, crowding formation, and scattering more precisely than conventional integer-order approaches. Six simulation scenarios demonstrate the fractional order effect and radius of nonlocal interaction of evolution-based traffic density. The key findings reflect that combining memory and nonlocality particularly enhances the capabilities of predictions and realism. The proposed framework offers a basis for modern traffic control schemes and opens novel trends for stratifying fractional PDEs in detailed dynamical systems.