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Conservation laws and some new exact solutions for traffic flow model via symmetry analysis

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  • Shagolshem, Sumanta
  • Bira, B.
  • Sil, Subhankar

Abstract

In this paper, we investigate the traffic flow model with congested phase through local and nonlocal symmetry analysis. Firstly, we derive the Lie group of transformations and show that it admits four one-dimensional optimal algebras. Next, by similarity reductions, we construct several exact solutions for each subalgebras as well as analyze the relation of group parameters. Furthermore, we construct a tree representing nonlocally related partial differential equations (PDEs) consisting inverse potential systems (IPS) and potential systems. Then, we prove that the traffic flow model yields one nonlocal symmetry and hence we derive an exact solution for the given system. Moreover, we generate the conservation laws for the governing systems through the nonlinear self-adjoint property. Finally, we study the nonlinear behaviors like weak discontinuity (C1-wave), characteristic shock for the physical model, and the impact of the anticipation factor on their evolutionary behavior graphically.

Suggested Citation

  • Shagolshem, Sumanta & Bira, B. & Sil, Subhankar, 2022. "Conservation laws and some new exact solutions for traffic flow model via symmetry analysis," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    • Handle: RePEc:eee:chsofr:v:165:y:2022:i:p1:s0960077922009584
    DOI: 10.1016/j.chaos.2022.112779
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    References listed on IDEAS

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    1. Paul I. Richards, 1956. "Shock Waves on the Highway," Operations Research, INFORMS, vol. 4(1), pages 42-51, February.
    2. Satapathy, Purnima & Raja Sekhar, T., 2018. "Optimal system, invariant solutions and evolution of weak discontinuity for isentropic drift flux model," Applied Mathematics and Computation, Elsevier, vol. 334(C), pages 107-116.
    3. Sil, Subhankar & Raja Sekhar, T. & Zeidan, Dia, 2020. "Nonlocal conservation laws, nonlocal symmetries and exact solutions of an integrable soliton equation," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    4. Nagel, Kai & Herrmann, Hans J., 1993. "Deterministic models for traffic jams," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 199(2), pages 254-269.
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    Cited by:

    1. Rosa, M. & Gandarias, M.L. & Niño-López, A. & Chulián, S., 2023. "Exact solutions through symmetry reductions for a high-grade brain tumor model with response to hypoxia," Chaos, Solitons & Fractals, Elsevier, vol. 171(C).

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