2019/02/22 by Jereme Chien, Wen Shen, Chien, Jereme +1
Engineering · Mathematics · #34B99 #35L02 #35L65 #35Q99 #65M20 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Differential Equations and Numerical Methods #FOS: Mathematics #Fractional Differential Equations Solutions #Traffic control and management
paper · pdf · doi:10.48550/arxiv.1902.08537
openalex publication_date 2019/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a nonlocal particle model describing traffic flow on rough roads. In the model, each driver adjusts the speed of the car according to the condition over an interval in the front, leading to a system of nonlocal ODEs which we refer to as the FtLs (follow-the-leaders) model. Assuming that the road condition is discontinuous at the origin, we seek stationary wave profiles (see Definition 1.1) for the system of ODEs across this discontinuity. We derive a non-local delay differential equation with discontinuous coefficient, satisfied by the profiles, together with conditions on the asymptotic values as x→±∞. Results on existence, uniqueness, and local stability are proved, for all cases. We show that, depending on the case, there might exist a unique profile, infinitely many profiles, or no profiles. The stability result also depends on cases. Various numerical simulations are presented. Finally, we establish convergence of these profiles to those of a local particle model, as well as those of a nonlocal PDE model.