Fractal Dynamical Model of Vehicular Traffic Flow within the Local Fractional Conservation Laws
We suggest a new model of the scale conservation equation in the mathematical theory of vehicular traffic flow on the fractal network based on the local fractional calculus.
Continuum model was applied to describe the traffic flow with the continuous functions which was analogous to one of fluid dynamics and material models based on the conservation laws. The approach started with Lighthill, Whitham, and Richards’s (LWR) works [1, 2]. The LWR model was studied by many authors, for example, Daganzo , Zhang , Li , Gasser , Aw et al. , Bellomo and Coscia , and Bellomo et al. . The LWR model of kinematic traffic waves derived from the conservation laws was [1–9] where the quantity is the density of time and space and the quantity is the vehicle flux as a function of density and speed with , and its solution for the equation was discussed by using the finite difference method .
Starting with Mandelbrot  there were many reports to determine the fractal structure of nature in various fields of science and engineering. We recall that the geometric similarity of traffic networks was reported by many researchers, for example, Erramilli et al. , Lam and Wornell , Shang et al. , Li et al. , and Campari and Levi . Recently, the local fractional calculus suggested in  was applied to deal with the nondifferentiable phenomena [18–22]. For example, local fractional Navier-Stokes equations were suggested in  and the local fractional Helmholtz and diffusion equations were reported in . The local fractional Maxwell’s equations were proposed in . Local fractional nonhomogeneous heat equations were investigated in . The heat transfer in silk cocoon hierarchy with local fractional derivative was proposed in . When the physical quantity of density or speed (denoted in Figure 1 by ) for vehicular traffic flow on the fractal network is a nondifferentiable function with time and space defined on Cantor sets, the classical conservation law is no valid.
In order to overcome the above drawback, in this paper we discuss the fractal dynamical model of vehicular traffic flow within the local fractional conservation laws. The outline of the paper is as follows. In Section 2, we recall the local fractional conservation laws. In Section 4, a mathematical model of vehicular traffic flow with fractal network is suggested. In Section 5, the nonhomogeneous partial differential equations for the vehicular traffic flow with fractal network are discussed. The conclusions are shown in Section 5.
2. Local Fractional Conservation Laws
In this section, we introduce the local fractional conservation laws based on the local fractional calculus. We start with the conception of the local fractional vector integrals used in the paper.
Definition 1. The local fractional surface integral is defined as given below [17–21]: where the quantity is the elements of fractal surface, is the elements of area with a unit normal local fractional vector , and as .
For any arbitrary volume, from (6) we obtain the local fractional differential form of the local fractional conservation laws as
In view of (8), the local fractional conservation law in the direction reads as follows: where the fractal flux denotes and the local fractional partial derivative of of order is defined as  where
We recall the local fractional conservation laws, which had been successfully applied to deal with elasticity , fluid mechanics , diffusion , electromagnetic  and heat  flows, and so on.
3. Fractal Dynamical Model of Vehicular Traffic Flow with Network
In this section, we study the fractal dynamical model of vehicular traffic flow with network. In order to derive it, we consider that the number of vehicles on the fractal homogeneous road without sources and sinks is always conserved and that fractal flow is a product of the density and speed, which is a differentiable function. We now start with the derivation of the local fractional conservative law shown in Figure 2.
The local fractional integral of of order in the interval is given by  where the partitions of the interval are denoted as , , , and with and . The number of the vehicular traffics in the segment at time is which leads to where the density of traffic flow is and the quantities of traffic flux in fixed time are
Let us consider
In view of (14) and (16), we have therefore, where and are arbitrary. Equation (18) represents the Lighthill-Whitham-Richards model of fractal traffic flow with local fractional derivative. We notice that (18) is in agreement with (9).
For the traffic flow , (18) can be written as with the initial value condition
From (24), we easily obtain the Lighthill-Whitham-Richards model on a finite length highway with the initial and boundary conditions
This equation is the liner Lighthill-Whitham-Richards model of fractal traffic flow with local fractional derivative.
Let us consider a linear velocity given as then, the expression of the traffic flow becomes where is the fractal unimpeded traffic speed and represents the maximum density.
From (19) and (29) we obtain the nonlinear local fractional partial differential equation where is density of fractal traffic flow. This equation is the nonlinear Lighthill-Whitham-Richards model of fractal traffic flow with local fractional derivative.
From (30) we derive Cauchy problem of the nonlinear Lighthill-Whitham-Richards model of fractal traffic flow such that
In this section, we investigate the nonhomogeneous partial differential equations for the vehicular traffic flow with fractal network.
From (24) and (25), the Cauchy problem of the nonhomogeneous partial differential equation of Lighthill-Whitham-Richards model with nondifferentiable source term is given by subject to the initial value condition
In view of (26) and (27), the initial and boundary problems for the nonhomogeneous partial differential equation of Lighthill-Whitham-Richards model with nondifferentiable source term on a finite length highway become with the initial and boundary conditions
Considering (31) and (32), the Cauchy problem of the nonhomogeneous nonlinear partial differential equation of Lighthill-Whitham-Richards model with nondifferentiable source term becomes with the initial value condition
By taking into account (33), the initial-boundary problem for nonhomogeneous nonlinear partial differential equation of Lighthill-Whitham-Richards model with nondifferentiable source term can be written as subjected to with , for , .
In this work, the fractal dynamical models of vehicular traffic flow within the local fractional conservation laws, where the density and speed of fractal traffic flow are nondifferentiable functions, are investigated. Besides, the linear and nonlinear partial differential equations for Lighthill-Whitham-Richards models of the vehicular traffic flows with fractal networks are obtained. The classical results are special case of the ones with nondifferentiable conditions when the fractal dimension is equal to 1.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Fundamental Research Funds for the Central Universities under Grant no. CHD2011JC191. It was also supported by the National Natural Science Foundation of China under Grant no. 51208054.
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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
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
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
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
B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, San Francisco, Calif, USA, 1982.View at: MathSciNet
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
X.-J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
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 Site | Google Scholar | MathSciNet
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 Site | Google Scholar
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