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

Said Agoujil; Abderrahman Bouhamidi; Sofiya Chergui; Youssef Qaraai

doi:10.3934/dcdss.2020022

Article Contents

2020, Volume 13, Issue 3: 389-406. Doi: 10.3934/dcdss.2020022

This issue Previous Article MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives Next Article New aspects of time fractional optimal control problems within operators with nonsingular kernel

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
^* Corresponding author: Sofiya Chergui

Received: June 30, 2018

Revised: July 31, 2018

Published: December 2019

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Abstract

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.

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Mathematics Subject Classification: 58J45, 35L02; Secondary: 35L65, 35L60.

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Figure 1. Approximate density (veh/m)

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Figure 2. Approximate density. Vs exact density (veh/m)

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Figure 3. The variations of Link occupancy in time

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Figure 4. The evolution of hydrocarbon emission rate in time

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Figure 5. Aggregate emission rate vs. link occupancy

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Figure 6. Flow.Vs density (fundamental diagram)

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Figure 7. Travel speed

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Table 1. 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}} $

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