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Abstract and Applied Analysis
Volume 2014, Article ID 635760, 5 pages
http://dx.doi.org/10.1155/2014/635760
Research Article

Fractal Dynamical Model of Vehicular Traffic Flow within the Local Fractional Conservation Laws

1School of Highway, Chang’an University, Xi’an 710064, China
2Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu 221008, China
3Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
4Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey
5Institute of Space Sciences, Magurele, Bucharest, Romania
6Department of Mathematics, University of Salerno, Via Ponte don Melillo, Fisciano, 84084 Salerno, Italy
7Electronic and Information Technology Department, Jiangmen Polytechnic, Jiangmen 529090, China

Received 25 February 2014; Accepted 7 March 2014; Published 1 April 2014

Academic Editor: Hari M. Srivastava

Copyright © 2014 Long-Fei Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. J. Lighthill and G. B. Whitham, “On kinematic waves—II. A theory of traffic flow on long crowded roads,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 229, no. 1178, pp. 317–345, 1955. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. P. I. Richards, “Shock waves on the highway,” Operations Research, vol. 4, no. 1, pp. 42–51, 1956. View at Publisher · View at Google Scholar · View at MathSciNet
  3. C. F. Daganzo, “A continuum theory of traffic dynamics for freeways with special lanes,” Transportation Research B: Methodological, vol. 31, no. 2, pp. 83–102, 1997. View at Google Scholar · View at Scopus
  4. H. M. Zhang, “New perspectives on continuum traffic flow models,” Networks and Spatial Economics, vol. 1, no. 1-2, pp. 9–33, 2001. View at Google Scholar
  5. T. Li, “L1 stability of conservation laws for a traffic flow model,” Electronic Journal of Differential Equations, vol. 2001, no. 14, pp. 1–18, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. I. Gasser, “On non-entropy solutions of scalar conservation laws for traffic flow,” Journal of Applied Mathematics and Mechanics, vol. 83, no. 2, pp. 137–143, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. Aw, A. Klar, T. Materne, and M. Rascle, “Derivation of continuum traffic flow models from microscopic follow-the-leader models,” SIAM Journal on Applied Mathematics, vol. 63, no. 1, pp. 259–278, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. N. Bellomo and V. Coscia, “First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow,” Comptes Rendus Mecanique, vol. 333, no. 11, pp. 843–851, 2005. View at Publisher · View at Google Scholar · View at Scopus
  9. N. Bellomo, M. Delitala, and V. Coscia, “On the mathematical theory of vehicular traffic flow—I: fluid dynamic and kinetic modelling,” Mathematical Models and Methods in Applied Sciences, vol. 12, no. 12, pp. 1801–1843, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. C. F. Daganzo, “A finite difference approximation of the kinematic wave model of traffic flow,” Transportation Research B: Methodological, vol. 29, no. 4, pp. 261–276, 1995. View at Google Scholar · View at Scopus
  11. B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, San Francisco, Calif, USA, 1982. View at MathSciNet
  12. A. Erramilli, W. Willinger, and P. Pruthi, “Fractal traffic flows in high-speed communications networks,” Fractals, vol. 2, no. 3, pp. 409–412, 1994. View at Google Scholar
  13. W. M. Lam and G. W. Wornell, “Multiscale analysis and control of networks with fractal traffic,” Applied and Computational Harmonic Analysis, vol. 11, no. 1, pp. 124–146, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. P. Shang, M. Wan, and S. Kama, “Fractal nature of highway traffic data,” Computers & Mathematics with Applications, vol. 54, no. 1, pp. 107–116, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. Li, W. Zhao, and C. Cattani, “Delay bound: fractal traffic passes through network servers,” Mathematical Problems in Engineering, vol. 2013, Article ID 157636, 15 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  16. E. G. Campari and G. Levi, “Self-similarity in highway traffic,” The European Physical Journal B: Condensed Matter and Complex Systems, vol. 25, no. 2, pp. 245–251, 2002. View at Publisher · View at Google Scholar
  17. X.-J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
  18. X.-J. Yang, D. Baleanu, and J. A. T. Machado, “Systems of Navier-Stokes equations on Cantor sets,” Mathematical Problems in Engineering, vol. 2013, Article ID 769724, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Y.-J. Hao, H. M. Srivastava, H. Jafari, and X.-J. Yang, “Helmholtz and diffusion equations associated with local fractional derivative operators involving the Cantorian and Cantor-type cylindrical coordinates,” Advances in Mathematical Physics, vol. 2013, Article ID 754248, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  20. Y. Zhao, D. Baleanu, C. Cattani, D.-F. Cheng, and X.-J. Yang, “Maxwell's equations on Cantor sets: a local fractional approach,” Advances in High Energy Physics, vol. 2013, Article ID 686371, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  21. A.-M. Yang, C. Cattani, C. Zhang, G.-N. Xie, and X.-J. Yang, “Local fractional Fourier series solutions for non-homogeneous heat equations arising in fractal heat flow with local fractional derivative,” Advances in Mechanical Engineering, vol. 2014, Article ID 514639, 5 pages, 2014. View at Publisher · View at Google Scholar
  22. J.-H. He and F.-J. Liu, “Local fractional variational iteration method for fractal heat transfer in silk cocoon hierarchy,” Nonlinear Science Letters A, vol. 4, no. 1, pp. 15–20, 2013. View at Google Scholar