The time fractional order derivative for multi-class AR model

TL;DR

This paper introduces a multi-class Aw-Rascle traffic flow model incorporating time fractional derivatives, providing a novel approach to analyze vehicle class interactions with a stable numerical scheme.

Contribution

It presents the first multi-class AR model with time fractional derivatives, along with a consistent, stable, and convergent finite difference scheme for its numerical solution.

Findings

Fractional order derivative influences traffic flow dynamics.

Numerical scheme is stable and convergent.

Results are realistic within model limits.

Abstract

In this paper, a multi-class Aw-Rascle \textrm{(AR)} model with time fractional order derivative is presented. The conservative form of the proposed model is considered for the natural extension and generalization of equations involved. The fractional order derivative involved in the model equations is computed by applying the Caputo fractional derivative definition. An explicit difference scheme is obtained through finite difference method of discretization. The scheme is shown to be consistent, conditionally stable and convergent. Numerical flux is computed by original Roe decomposition and an entropy condition applied to the Roe decomposition. From numerical results, the effect of fractional-order derivative of time, on the traffic flow of vehicle classes is determined. Results obtained from the proposed model remain within limits therefore, they are realistic.