On the solution of two-dimensional local fractional LWR model of fractal vehicular traffic flow

Document Type : Original Article

Authors

School of Liberal Studies, Ambedkar University Delhi, India

Abstract
This study proposed a two-dimensional local fractional Lighthill-Whitham-Richards (LWR) model of fractal vehicular traffic flow. The non-differentiable traffic parameters that arise in traffic flow are addressed by this model. The local fractional Laplace variational iteration method (LFLVIM) is employed to analyse the proposed model and making it serves well for examining the dynamic shifts in non-differentiable traffic density function. The existence and uniqueness of the solution of 2D local fractional LWR model have also proven. Several examples are also covered to further illustrate the effectiveness of applying LFLVIM to the proposed model. Additionally, the numerical simulations for every instance have been demonstrated. This study indicates that the presented 2D fractal model well captures the phenomena of traffic flow, and the iterative approach that has been employed to analyse the model is successful and can be applied to obtain the non-differentiable solution to 2D local fractional LWR model.

Keywords

Subjects


[1] Agrawal, S., Kanagaraj, V., & Treiber, M. (2023). Two-dimensional LWR model for lane-free traffic. Physica A: Statistical Mechanics and its Applications, 625, 128990. 1, 3, 6
[2] Amin, S., Andrews, S., Apte, S., Arnold, J., Ban, J., Benko, M., ... & Tinka, A. (2008, November). Mobile century using GPS mobile phones as traffic sensors: a field experiment. In 15th World Congress on Intelligent Transportation Systems (Vol. 2008, p. 18). IEEE. 1
[3] Aw, A. A. T. M., & Rascle, M. (2000). Resurrection of" second order" models of traffic flow. SIAM journal on applied mathematics, 60(3), 916-938. 1, 1
[4] Babakhani, A., & Daftardar-Gejji, V. (2002). On calculus of local fractional derivatives. Journal of Mathematical Analysis and Applications, 270(1), 66-79. 1
[5] Baleanu, D., & Jassim, H. K. (2019). A modification fractional homotopy perturbation method for solving Helmholtz and coupled Helmholtz equations on Cantor sets. Fractal and Fractional, 3(2), 30. 1
[6] Baleanu, D., Jassim, H. K., & Al Qurashi, M. (2019). Solving Helmholtz equation with local fractional derivative operators. Fractal and Fractional, 3(3), 43. 1
[7] Balzotti, C., & Göttlich, S. (2020). A two-dimensional multi-class traffic flow model. arXiv preprint arXiv:2006.10131. 1
[8] Blandin, S., Bretti, G., Cutolo, A., & Piccoli, B. (2009). Numerical simulations of traffic data via fluid dynamic approach. Applied Mathematics and Computation, 210(2), 441-454. 1
[9] Cannon, J. W. (1984). The fractal geometry of nature. by Benoit B. Mandelbrot. The American Mathematical Monthly, 91(9), 594-598. 1
[10] Chen, W., Sun, H., Zhang, X., & Koroak, D. (2010). Anomalous diffusion modeling by fractal and fractional derivatives. Computers & Mathematics with Applications, 59(5), 1754-1758. 1
[11] Claudel, C. G., & Bayen, A. M. (2010). LaxHopf based incorporation of internal boundary conditions into Hamil- tonJacobi equation. Part I: Theory. IEEE Transactions on Automatic Control, 55(5), 1142-1157. 1
[12] Cui, P., & Jassim, H. K. (2024). Local fractional Sumudu decomposition method to solve fractal PDEs arising in mathematical physics. Fractals, 32(04), 2440029. 1
[13] Daganzo, C. F. (2002). A behavioral theory of multi-lane traffic flow. Part I: Long homogeneous freeway sections. Transportation Research Part B: Methodological, 36(2), 131-158. 1
[14] Fan, S., Herty, M., & Seibold, B. (2013). Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model. arXiv preprint arXiv:1310.8219. 1
[15] Gazis, D. C., Herman, R., & Weiss, G. H. (1962). Density oscillations between lanes of a multilane highway. Operations Research, 10(5), 658-667. 1
[16] Gupta, A. K., & Dhiman, I. (2014). Analyses of a continuum traffic flow model for a nonlane-based system. International Journal of Modern Physics C, 25(10), 1450045. 1
[17] Hao, Y. J., Srivastava, H. M., Jafari, H., & Yang, X. J. (2013). Helmholtz and diffusion equations associated with local fractional derivative operators involving the Cantorian and Cantortype cylindrical coordinates. Advances in Mathematical Physics, 2013(1), 754248. 1
[18] He, J. H., Elagan, S. K., & Li, Z. B. (2012). Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Physics letters A, 376(4), 257-259. 1
[19] Helbing, D., & Treiber, M. (1999). Numerical simulation of macroscopic traffic equations. Computing in Science & Engineering, 1(5), 89-98. 1
[20] Herty, M., Fazekas, A., & Visconti, G. (2017). A two-dimensional data-driven model for traffic flow on highways. arXiv preprint arXiv:1706.07965. 1
[21] Jafari, H., & Jassim, H. K. (2015). Local fractional variational iteration method for solving nonlinear partial dif-
ferential equations within local fractional operators. Applications and Applied Mathematics: An International Journal (AAM), 10(2), 29. 2.4, 2.5
[22] Jafari, H., Jassim, H. K., Ünlü, C., & Nguyen, V. T. (2024). Laplace decomposition method for solving the two- dimensional diffusion problem in fractal heat transfer. Fractals, 32(04), 2440026. 1
[23] Jafari, H., Jassim, H. K., Ansari, A., & Nguyen, V. T. (2024). Local fractional variational iteration transform method: A tool for solving local fractional partial differential equations. Fractals, 32(04), 2440022. 1
[24] Jafari, H., Zair, M. Y., & Jassim, H. K. (2023). Analysis of fractional NavierStokes equations. Heat Transfer, 52(3), 2859-2877. 1
[25] Jassim, H. K., & Khafif, S. A. (2021). SVIM for solving Burgers and coupled Burgers equations of fractional order. Progress in Fractional Differentiation and Applications, 7(1), 1-6. 1
[26] Kamil Jassim, H., & Vahidi, J. (2021). A new technique of reduce differential transform method to solve local
fractional PDEs in mathematical physics. International Journal of Nonlinear Analysis and Applications, 12(1), 37-44. 1
[27] Laval, J. A., & Daganzo, C. F. (2006). Lane-changing in traffic streams. Transportation Research Part B: Methodological, 40(3), 251-264. 1
[28] Lebacque, J. P. (1996, July). The godunov scheme and what it means for rst order tra c ow models. In Proceedings of the 13th International Symposium on Transportation and Traffic Theory, Lyon, France, July (Vol. 2426). 1
[29] Liang, Y., Allen, Q. Y., Chen, W., Gatto, R. G., Colon-Perez, L., Mareci, T. H., & Magin, R. L. (2016). A fractal
derivative model for the characterization of anomalous diffusion in magnetic resonance imaging. Communications in Nonlinear Science and Numerical Simulation, 39, 529-537. 1
[30] Liang, Y., Chen, W., Xu, W., & Sun, H. (2019). Distributed order Hausdorff derivative diffusion model to charac-
terize non-Fickian diffusion in porous media. Communications in Nonlinear Science and Numerical Simulation, 70, 384-393. 1
[31] Lighthill, M. H., Whitham, G. B. (1961). II-A Theory of Traffic Flow on Long Crowded Roads. Proceedings of the Royal Society of London: Mathematical and physical sciences, 229, 317. 1
[32] Liu, J. G., Yang, X. J., Feng, Y. Y., & Cui, P. (2020). A new perspective to study the third-order modified KDV equation on fractal set. Fractals, 28(06), 2050110. 1
[33] Mohan, R., & Ramadurai, G. (2021). Multi-class traffic flow model based on three dimensional flowconcentration surface. Physica A: Statistical Mechanics and its Applications, 577, 126060. 1
[34] Munjal, P. K., & Pipes, L. A. (1971). Propagation of on-ramp density waves on uniform unidirectional multilane freeways. Transportation science, 5(4), 390-402. 1
[35] Richards, P. I. (1956). Shock waves on the highway. Operations research, 4(1), 42-51. 1
[36] Vikram, D., Mittal, S., & Chakroborty, P. (2022). Stabilized finite element computations with a two-dimensional continuum model for disorderly traffic flow. Computers & Fluids, 232, 105205. 1
[37] Wang, L. F., Yang, X. J., Baleanu, D., Cattani, C., & Zhao, Y. (2014). Fractal dynamical model of vehicular traffic flow within the local fractional conservation laws. In Abstract and Applied Analysis (Vol. 2014, No. 1, p. 635760). Hindawi Publishing Corporation. 1
[38] Yang, A. M., Cattani, C., Zhang, C., Xie, G. N., & Yang, X. J. (2014). Local fractional Fourier series solutions for nonhomogeneous heat equations arising in fractal heat flow with local fractional derivative. Advances in Mechanical Engineering, 6, 514639. 1, 1, 2.8, 2.9
[39] Yang, X. J. (2012). Advanced local fractional calculus and its applications. 1, 2.1, 2.2, 2.3, 2.6, 2.7, 5
[40] Yang, X. J., Baleanu, D., & Tenreiro Machado, J. A. (2013). Systems of NavierStokes equations on Cantor sets. Mathematical Problems in Engineering, 2013(1), 769724. 1, 1
[41] Yang, X. J. (2011). Local Fractional Functional Analysis & Its Applications (Vol. 1). Hong Kong: Asian Academic Publisher Limited. 1, 2.8, 2.9
[42] Zayir, M. Y., & Kamil Jassim, H. (2022). A Fractional Variational Iteration Approach for Solving Time-Fractional Navier-Stokes Equations. Mathematics and Computational Sciences, 3(2), 41-47. 1
[43] Zhang, H. M. (2002). A non-equilibrium traffic model devoid of gas-like behavior. Transportation Research Part B: Methodological, 36(3), 275-290. 1
[44] Zhao, Y., Baleanu, D., Cattani, C., Cheng, D. F., & Yang, X. J. (2013). Maxwells equations on Cantor sets: a local fractional approach. Advances in high energy physics, 2013(1), 686371. 1, 1
Volume 6, Issue 2
Spring 2025
Pages 43-63

  • Receive Date 23 September 2024
  • Revise Date 04 February 2025
  • Accept Date 05 February 2025