Research Article | Open Access

Yang Li, Long-Fei Wang, Sheng-Da Zeng, Yang Zhao, "Local Fractional Laplace Variational Iteration Method for Fractal Vehicular Traffic Flow", *Advances in Mathematical Physics*, vol. 2014, Article ID 649318, 7 pages, 2014. https://doi.org/10.1155/2014/649318

# Local Fractional Laplace Variational Iteration Method for Fractal Vehicular Traffic Flow

**Academic Editor:**Xiao-Jun Yang

#### Abstract

We discuss the line partial differential equations arising in fractal vehicular traffic flow. The nondifferentiable approximate solutions are obtained by using the local fractional Laplace variational iteration method, which is the coupling method of local fractional variational iteration method and Laplace transform. The obtained results show the efficiency and accuracy of implements of the present method.

#### 1. Introduction

Fractional differential equations with arbitrary orders were applied to model the real-world problems for science and engineering. Many researchers present their applications with solid mechanics, heat transfer, fluid mechanics, transport process, water motion, and quantum mechanics. For example, Tarasov studied the wave equation for fractal solid string based on the fractional calculus [1]. Momani and Odibat presented the linear, nonlinear, and fractional partial differential equations arising in fluid mechanics [2, 3]. Povstenko suggested the fractional heat conduction equation [4], and Vázquez gave the second law of thermodynamics with fractional derivative [5]. Lutz considered the fractional transport equation for Levy stable processes [6], and Kadem et al. discussed its solution using the spectral method [7]. Laskin presented the fractional Schrodinger equation [8], and Muslih et al. suggested its solution [9]. There are many methods for finding the solutions to the fractional differentiable equations [10]. For example, Jafari and Seifi used the homotopy analysis method to deal with linear and nonlinear fractional diffusion-wave equations [11]. Jafari et al. applied the fractional subequation method to solve the Cahn-Hilliard and Klein-Gordon equations [12]. Hristov suggested the heat-balance integral method for fractional heat diffusion and transient flow [13, 14]. Bhrawy and Alghamdi proposed the shifted Jacobi-Gauss-Lobatto collocation method to find the solution for nonlinear fractional Langevin equation with two variables [15] and the shifted Legendre spectral method for solving the fractional-order multipoint boundary value problems [16]. Bhrawy and Baleanu considered the spectral Legendre-Gauss-Lobatto collocation method to solve the space-fractional advection diffusion equations [17]. Atangana and Baleanu presented the two difference methods to solve the fractional parabolic equations [18].

Recently, the local fractional calculus is proposed and developed to describe the fractal problems in various fields, such as physics [19–21], applied mathematics [22, 23], signal processing [24–27], fluid mechanics [28], quantum mechanics [29], fractal forest gap [30], vehicular traffic flow [31], and silk cocoon hierarchy [32]. The linear differential equation arising in fractal vehicular traffic flow was suggested in [31]. The local fractional Laplace variational iteration method was suggested in [23] and developed in [33]. In this paper, we use the local fractional Laplace variational iteration method to solve the linear differential equation arising in fractal vehicular traffic flow. The structure of the paper is suggested as follows. In Section 2, the basic theory of local fractional calculus and local fractional Laplace transform are introduced. Section 3 gives the local fractional Laplace variational iteration method. In Section 4, the nondifferentiable solutions for line partial differential equations arising in fractal vehicular traffic flow are presented. Finally, the conclusions are considered in Section 5.

#### 2. The Lighthill-Whitham-Richards Model on a Finite Length Highway

In this section, we present the Lighthill-Whitham-Richards model on a finite length highway, the conceptions of local fractional derivative and integral, and the local fractional Laplace transform.

The Lighthill-Whitham-Richards model on a finite length highway reads as follows [31]: where the initial and boundary conditions are presented as follows: with for , for , , and the local fractional partial derivative of of order is given as [19] with This is the line partial differential equation arising in fractal vehicular traffic flow.

The local fractional derivative of of order is expressed as [19, 20] where

The local fractional integral of of order in the interval is defined through [19, 21, 22] where the partitions of the interval are , with , , , and , .

The local fractional Laplace transform is given as [23, 26, 33–35] where is local fractional continuous and .

The inverse formula of local fractional Laplace transform is suggested as [23, 26, 33–35] where and .

The local fractional convolution of two functions is presented as [23, 33] The properties for local fractional Laplace transform are listed as follows [23, 26, 33–35]:

#### 3. Local Fractional Laplace Variational Iteration Method

In this section, we introduce the local fractional Laplace variational iteration method. Let us consider the following local fractional differential operator: where the linear local fractional differential operator is and is a nondifferential function.

According to the local fractional Laplace variational iteration method [23, 33], the local fractional functional formula is presented as follows: Applying the local fractional Laplace transform gives the following: Taking the local fractional variation of (15), we obtain From (16), we have such that where Therefore, we get Using formula (20), we arrive at new iteration algorithm as follows: where the initial value reads as Thus, the local fractional series solution of (13) is which leads to

#### 4. The Nondifferentiable Solutions for Line Partial Differential Equations Arising in Fractal Vehicular Traffic Flow

In this section, we present the boundary value problems for line partial differential equations arising in fractal vehicular traffic flow.

*Example 1. *The initial and boundary conditions for line partial differential equations arising in fractal vehicular traffic flow read as follows:
In view of (21), we have
where the initial value is
Making use of (26) and (27), we have the first approximation
In view of (26) and (28), we arrive at the second approximation
From (26) and (29), the third approximation is
Applying (26) and (29) gives the fourth approximation
In view of (26) and (31), we give the fifth approximation
Therefore, we obtain
which reduces to
and its graph is shown in Figure 1 with the parameters and .

*Example 2. *The initial and boundary conditions for line partial differential equations arising in fractal vehicular traffic flow with the parameter are
From (21), we obtain
where the initial value is
In view of (36) and (37), we get the first approximation
From (36) and (38), we obtain the second approximation as follows:
Using (36) and (39), we have the third approximation
Making use of (36) and (40), we have the fourth approximation
From (36) and (41), we get the fifth approximation as follows:
Thus, we obtain the local fractional series as follows:
which yields
and its graph is shown in Figure 2.

#### 5. Conclusions

In this paper, the boundary value problems for line partial differential equations arising in fractal vehicular traffic flow were solved by using the local fractional Laplace variational iteration method, which is the coupling method of local fractional variational iteration method (a generalization of variational iteration method based upon the local fractional calculus) and Laplace transform (a generalization of Fourier transform based upon the local fractional calculus). The nondifferentiable approximate solutions were obtained and their graphs were also shown.

#### Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

#### References

- V. E. Tarasov, “Wave equation for fractal solid string,”
*Modern Physics Letters B*, vol. 19, no. 15, pp. 721–728, 2005. View at: Google Scholar - S. Momani and Z. Odibat, “Analytical approach to linear fractional partial differential equations arising in fluid mechanics,”
*Physics Letters A: General, Atomic and Solid State Physics*, vol. 355, no. 4-5, pp. 271–279, 2006. View at: Publisher Site | Google Scholar - S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,”
*Applied Mathematics and Computation*, vol. 177, no. 2, pp. 488–494, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Z. Povstenko, “Fractional heat conduction equation and associated thermal stress,”
*Journal of Thermal Stresses*, vol. 28, no. 1, pp. 83–102, 2005. View at: Publisher Site | Google Scholar | MathSciNet - L. Vázquez, J. J. Trujillo, and M. P. Velasco, “Fractional heat equation and the second law of thermodynamics,”
*Fractional Calculus and Applied Analysis*, vol. 14, no. 3, pp. 334–342, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. Lutz, “Fractional transport equations for Lévy stable processes,”
*Physical Review Letters*, vol. 86, no. 11, pp. 2208–2211, 2001. View at: Publisher Site | Google Scholar - A. Kadem, Y. Luchko, and D. Baleanu, “Spectral method for solution of the fractional transport equation,”
*Reports on Mathematical Physics*, vol. 66, no. 1, pp. 103–115, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - N. Laskin, “Fractional Schrödinger equation,”
*Physical Review E. Statistical, Nonlinear, and Soft Matter Physics*, vol. 66, no. 5, Article ID 056108, 7 pages, 2002. View at: Publisher Site | Google Scholar | MathSciNet - S. I. Muslih, O. P. Agrawal, and D. Baleanu, “A fractional Schrödinger equation and its solution,”
*International Journal of Theoretical Physics*, vol. 49, no. 8, pp. 1746–1752, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, vol. 3 of*Complexity, Nonlinearity and Chaos*, World Scientific, Boston, Mass, USA, 2012. - H. Jafari and S. Seifi, “Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 5, pp. 2006–2012, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Jafari, H. Tajadodi, N. Kadkhoda, and D. Baleanu, “Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 587179, 5 pages, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Hristov, “Heat-balance integral to fractional (half-time) heat diffusion sub-model,”
*Thermal Science*, vol. 14, no. 2, pp. 291–316, 2010. View at: Publisher Site | Google Scholar - J. Hristov, “Transient flow of a generalized second grade fluid due to a constant surface shear stress: an approximate integral-balance solution,”
*International Review of Chemical Engineering*, vol. 3, no. 6, pp. 802–809, 2011. View at: Google Scholar - A. H. Bhrawy and M. A. Alghamdi, “A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals,”
*Boundary Value Problems*, vol. 2012, article 62, 2012. View at: Publisher Site | Google Scholar - A. H. Bhrawy and M. M. Al-Shomrani, “A shifted Legendre spectral method for fractional-order multi-point boundary value problems,”
*Advances in Difference Equations*, vol. 2012, article 8, 2012. View at: Publisher Site | Google Scholar - A. H. Bhrawy and D. Baleanu, “A spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients,”
*Reports on Mathematical Physics*, vol. 72, no. 2, pp. 219–233, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Atangana and D. Baleanu, “Numerical solution of a kind of fractional parabolic equations via two difference schemes,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 828764, 8 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - X.-J. Yang,
*Advanced Local Fractional Calculus and Its Applications*, World Science, New York, NY, USA, 2012. - X.-J. Yang, H. M. Srivastava, J.-H. He, and D. Baleanu, “Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives,”
*Physics Letters A*, vol. 377, no. 28–30, pp. 1696–1700, 2013. View at: Publisher Site | Google Scholar | MathSciNet - 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 Site | Google Scholar | MathSciNet - X.-J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,”
*Thermal Science*, vol. 17, no. 2, pp. 625–628, 2013. View at: Publisher Site | Google Scholar - C. F. Liu, S. S. Kong, and S. J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,”
*Thermal Science*, vol. 17, no. 3, pp. 715–721, 2013. View at: Google Scholar - X. J. Yang, D. Baleanu, and J. T. Machado, “Application of the local fractional Fourier series to fractal signals,” in
*Discontinuity and Complexity in Nonlinear Physical Systems*, pp. 63–89, Springer, 2014. View at: Google Scholar - K. Liu, R.-J. Hu, C. Cattani, G.-N. Xie, X.-J. Yang, and Y. Zhao, “Local fractional $Z$-transforms with applications to signals on cantor sets,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 638648, 6 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - G. Yi, “Local fractional Z transform in fractal space,”
*Advances in Digital Multimedia*, vol. 1, no. 2, pp. 96–102, 2012. View at: Google Scholar - X.-J. Yang, D. Baleanu, H. M. Srivastava, and J. A. Tenreiro Machado, “On local fractional continuous wavelet transform,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 725416, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - X.-J. Yang, D. Baleanu, and J. A. Tenreiro Machado, “Systems of Navier-Stokes equations on Cantor sets,”
*Mathematical Problems in Engineering*, vol. 2013, Article ID 769724, 8 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - X.-J. Yang, D. Baleanu, and J. A. Tenreiro Machado, “Mathematical aspects of Heisenberg uncertainty principle within local fractional Fourier analysis,”
*Boundary Value Problems*, vol. 2013, article 131, 2013. View at: Publisher Site | Google Scholar - C.-Y. Long, Y. Zhao, and H. Jafari, “Mathematical models arising in the fractal forest gap via local fractional calculus,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 782393, 6 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - L.-F. Wang, X.-J. Yang, D. Baleanu, C. Cattani, and Y. Zhao, “Fractal dynamical model of vehicular traffic flow within the local fractional conservation laws,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 635760, 5 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet - 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 - A. M. Yang, J. Li, H. M. Srivastava, G. N. Xie, and X. J. Yang, “The local fractional Laplace variational iteration method for solving linear partial differential equations with local fractional derivatives,”
*Discrete Dynamics in Nature and Society*, vol. 2014, Article ID 365981, 2014. View at: Publisher Site | Google Scholar - Y. Z. Zhang, A. M. Yang, and Y. Long, “Initial boundary value problem for fractal heat equation in the semi-infinite region by Yang-Laplace transform,”
*Thermal Science*, 2013. View at: Google Scholar - C.-G. Zhao, A.-M. Yang, H. Jafari, and A. Haghbin, “The Yang-Laplace transform for solving the IVPs with local fractional derivative,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 386459, 5 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Yang Li 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.