Selected mathematical problems of traffic flow theory

References

• Bando , M. , Hasabe , K. , Nakayama , A. , Shibata , A. and Sugiyama , Y. 1995 . Dynamical model of traffic congestion and numerical simulation . Phys. Rev. E , 51 ( 2 ) : 1035 – 1042 .
   PubMed Web of Science ®Google Scholar
• Belyaev , Y. K. 1969 . On simplest model of movement without overtaking . Izv. Akad. Nauk SSSR Tech. Kibern. , 3 : 17 – 21 . (in Russian)
   Google Scholar
• Blank , M. L. 2000 . Exact analyze of dynamic system applied in traffic flow models . Uspehi Matem. Nauk , 55 ( 3 ) : 167 – 168 . (in Russian)
   Google Scholar
• Blank , M. L. 2001 . Dynamics of traffic jams: Order and chaos . Mosc. Math. J. , 1 ( 1 ) : 1 – 26 .
   Web of Science ®Google Scholar
• Blank , M. 2003 . Ergodic properties of a simple deterministic traffic flow model . J. Stat. Phys. , 111 ( 3–4 ) : 903 – 930 . Available at arXiv:math.DS/0206194
   Google Scholar
• Blank , M. 2005 . Hysteresis phenomenon in deterministic traffic flows . J. Stat. Phys. , 120 ( 3–4 ) : 627 – 658 . Available at arXiv:math.DS/0408240
   Google Scholar
• Bugaev , A. S. , Buslaev , A. P. and Tatashev , A. G. 2006 . Monotonic random walk of particles on cellular lane and LYuMen problem . Math. Model. , 18 ( 12 ) : 19 – 34 .
   Google Scholar
• Bugaev , A. S. , Buslaev , A. P. and Tatashev , A. G. 2008 . Modeling of segregation of two-lane flow of particles . Math. Model. , 20 ( 9 ) : 111 – 119 .
   Google Scholar
• Buslaev , A. P. 2010 . Traffic flower with n petals . J. Appl. Funct. Anal. , 5 ( 1 ) : 85 – 99 .
   Google Scholar
• Buslaev , A. P. , Novikov , A. V. , Prikhodko , V. M. , Tatashev , A. G. and Yashina , M. V. 2003 . Stochastic and Simulation Approach to Traffic Optimization . Moscow, Mir , : 286 (in Russian)
   Google Scholar
• Buslaev , A. P. , Prikhodko , V. M. , Tatashev , A. G. and Yashina , M. V. 2005 . The deterministic-stochastic flow model . : 1 – 21 . Available at arXiv:physics/0504139
   Google Scholar
• Buslaev , A. P. , Gasnikov , A. V. and Yashina , M. V. Mathematical problems of traffic . Proceedings of the 10th International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE 2010), Almeria (Andalucia), Spain, June 26–30, 2010 . Edited by: Vigo Aguiar , J. Vol. 1 , pp. 307 – 313 .
   Google Scholar
• Chowdhury , D. , Santen , L. and Schadschneider , A. 2000 . Statistical physics of vehicular traffic and some related systems . Phys. Rep. , 329 ( 4–6 ) : 199 – 329 . arXiv:cond-mat/0007053v1
   Web of Science ®Google Scholar
• Dafermos , C. M. 2005 . “ Hyperbolic Conservation Laws in Continuum Physics ” . Berlin : Springer-Verlag .
   Google Scholar
• Daganzo , C. F. 1994 . The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory . Transp. Res. B , 28 ( 4 ) : 269 – 287 .
   Google Scholar
• Daganzo , C. F. 1995 . The cell transmission model, Part II: Network traffic . Transp. Res. B , 29 ( 2 ) : 79 – 93 .
   Google Scholar
• 1999 . “ Excellent Nature. A Book about Kolmogorov ” . 256 Moscow, Fasis, MIROS (in Russian)
   Google Scholar
• Florin , V. A. 1948 . Some of the simplest nonlinear problems arising in the consolidation of wet soil . Izv. Akad. Nauk SSSR Otdel. Tekhn. Nauk V. , 9 : 1389 – 1397 . (in Russian)
   Google Scholar
• Garavello , M. and Piccoli , B. 2006 . “ Traffic Flow on Networks ” . Springfield , , USA : American Institute of Mathematical Sciences . AIMS Series on Applied Mathematics, Vol. 1
   Google Scholar
• Gasnikov , A. V. 2009 . Time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity . Izv. RAN. Ser. Mat. , 73 ( 6 ) : 39 – 76 .
   Google Scholar
• Gelfand , I. M. 1959 . Some problems in the theory of quasilinear equations . UMN, , 14 ( 2 ) : 87 – 158 .
   Google Scholar
• Greenshields , B. D. 1934 . A study of traffic capacity . Proc. US Highway Res. Board , 14 : 448 – 494 .
   Google Scholar
• Helbing , D. 2001 . Traffic and related self-driven many particle systems . Rev. Mod. Phys. , 73 ( 4 ) : 1067 – 1141 . arXiv:cond-mat0012229
   Web of Science ®Google Scholar
• Henkin , G. M. 2007 . Asymptotic structure for solutions of the Cauchy problem for Burgers type equations . J. Fixed Point Theory Appl. , 1 ( 2 ) : 239 – 291 .
   Google Scholar
• Henkin , G. M. and Shananin , A. A. 2004 . Asymptotic behavior of solutions of the Cauchy problem for Burgers type equations . J. Math. Pures Appl. , 83 ( 12 ) : 1457 – 1500 .
   Google Scholar
• Holden , H. and Risebro , N. H. 2007 . “ Front Tracking for Hyperbolic Conservation Laws ” . Berlin : Springer-Verlag .
   Google Scholar
• Kerner , B. S. 2009 . “ Introduction to Modern Traffic Flow Theory and Control ” . Berlin : Springer .
   Google Scholar
• Kolmogorov , A. N. 1985 . Selected works. In 6 volumes. Volume 1: Mathematics and mechanics , 1 – 469 . Moscow : Nauka . (in Russian) (V.M. Tikhomirov and G.G. Magaril-Ilyaev, Inequalities for derivatives, pp. 387–389)
   Google Scholar
• Kruzhkov , S. N. 1970 . First order quasilinear equations with several independent variables . Mat. Sbornik , 81 ( 2 ) : 228 – 255 .
   Google Scholar
• Kurzhansky , A. A. 2007 . “ Modeling and software tools for freeway operational planning ” . Berkeley , CA : University of California . PhD thesis Available at http://lihodeev.com/pubs.html
   Google Scholar
• Lax , P. D. 2006 . “ Hyperbolic Partial Differential Equation ” . Providence , RI : American Mathematical Society .
   Google Scholar
• Lighthill , M. J. and Whitham , G. B. 1955 . On kinematic waves: II. Theory of traffic flow on long crowded roads . Proc. R. Soc. London, Ser. A , 229 : 281 – 345 .
   Web of Science ®Google Scholar
• Nagel , K. and Schreckenberg , M. 1992 . A cellular automation model for freeway traffic . J. Physique I , 2 : 2221 – 2229 .
   Google Scholar
• Newell , G. F. 1961 . Nonlinear effects in the dynamics of car – following . Oper. Res. , 9 : 209 – 229 .
   Web of Science ®Google Scholar
• Reuschel , A. 1950 . Fahrzeugbewegungen in der Kolonne . Österr. Ing., Arch. , 4 : 193 – 215 .
   Google Scholar
• Rothery , R. W. 1992 . “ Car following models ” . In Traffic Flow Theory, Transportation Research Board , Edited by: Gartner , N. , Messer , C. J. and Rathi , A. K. Vol. 165 , 1 – 41 . Washington : U.S. Department of Transportation . Chapter 4
   Google Scholar
• Treiber , M. , Hennecke , A. and Helbing , D. 2000 . Congested traffic states in empirical observations and microscopic simulation . Phys. Rev. E , 62 ( 2 ) : 1805 – 1824 .
   PubMed Web of Science ®Google Scholar
• Whitham , G. B. 1974 . “ Linear and Nonlinear Waves ” . New York : John Wiley & Sons .
   Google Scholar