# American Institute of Mathematical Sciences

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March  2020, 13(3): 389-406. doi: 10.3934/dcdss.2020022

## Implementation of the vehicular occupancy-emission relation using a cubic B-splines collocation method

 1 Department of Computer Sciences, Faculty of Sciences and Techniques, University Moulay Ismail, BP 509 Boutalamine Errachidia, Morocco 2 Department of Mathematics, Laboratory LMPA, University Littoral Cote d'Opale, France

* Corresponding author: Sofiya Chergui

Received  July 2018 Revised  August 2018 Published  March 2019

The complexity and non-linearity of flow phenomena are explained by numerous criteria, including the interactions of the large number of vehicles occupying the road, which influence the road density. This density under certain conditions, leads to traffic congestion which has dangerous effects on the environment such as; resources consumption; noise and the effect caused by greenhouse gas emissions of the $CO_{2}$ and other pollutants. In this paper we consider working in an uniform, homogeneous road where the traffic is described by the Lighthill Whitham-Richard (LWR) model resolved using a cubic B-spline collocation scheme in space and an implicit Runge Kutta scheme in time. We also shed light on the relation between vehicle occupancy and vehicle emissions.

Citation: Said Agoujil, Abderrahman Bouhamidi, Sofiya Chergui, Youssef Qaraai. Implementation of the vehicular occupancy-emission relation using a cubic B-splines collocation method. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 389-406. doi: 10.3934/dcdss.2020022
##### References:

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##### References:
Approximate density (veh/m)
Approximate density. Vs exact density (veh/m)
The variations of Link occupancy in time
The evolution of hydrocarbon emission rate in time
Aggregate emission rate vs. link occupancy
Flow.Vs density (fundamental diagram)
Travel speed
coefficients of $B_{j}$ and its derivatives
 $x$ $x_{j-1}$ $x_{j}$ $x_{j+1}$ $B_{j}$ $\frac{1}{6}$ $\frac{4}{6}$ $\frac{1}{6}$ $B^{'}_{j}$ $\frac{-1}{2h}$ $0$ $\frac{1}{2h}$ $B^{''}_{j}$ $\frac{1}{h^{2}}$ $\frac{-2}{h^{2}}$ $\frac{1}{h^{2}}$
 $x$ $x_{j-1}$ $x_{j}$ $x_{j+1}$ $B_{j}$ $\frac{1}{6}$ $\frac{4}{6}$ $\frac{1}{6}$ $B^{'}_{j}$ $\frac{-1}{2h}$ $0$ $\frac{1}{2h}$ $B^{''}_{j}$ $\frac{1}{h^{2}}$ $\frac{-2}{h^{2}}$ $\frac{1}{h^{2}}$
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