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Growth and decay of shock and acceleration waves in a traffic flow model with relaxation. (English) Zbl 1078.35073
This interesting paper is devoted to a traffic flow model with relaxation. The usual Ficken-based constitutive relation for traffic flux is replaced with one based on the Maxwell-Cattaneo model. This model leads to the nonlinear partial differential equation of hyperbolic type $\rho_t+\tau_0\rho_{tt}-\nu \rho_{xx}+v_{m}(1-2\rho /\rho_s )\rho_x=0 ,$ where $$\tau_0>0$$ is a relaxation (or lag) time, $$\rho (x,t) >0$$ is a density, the constant $$\rho_s$$ is the saturation value of the density ($$0<\rho <\rho_s$$), the constant $$\nu$$ is the diffusion coefficient (or diffusivity), the constant $$v_m>0$$ denotes the maximum speed as $$\rho \to 0$$, and $$x$$, $$t$$ denote the space and time, respectively. Here $$x$$ and/or $$t$$ subscripts denote partial differentiation. The exact traveling wave solution is derived. It is shown that shock formation is possible only if the diffusivity is non-vanishing. Exact amplitude expressions for both shock and acceleration waves are obtained as well as their temporal evolution is determined. The exact upper bound of the reaction time parameter is established. The connection between singular surface and traveling wave solution is discussed as well.

##### MSC:
 35L67 Shocks and singularities for hyperbolic equations 35L70 Second-order nonlinear hyperbolic equations 90B20 Traffic problems in operations research
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##### References:
 [1] Murray, J.D., Mathematical biology, (1993), Springer New York · Zbl 0779.92001 [2] Whitham, G.B., Linear and nonlinear waves, (1974), Wiley New York · Zbl 0373.76001 [3] Bland, D.R., Wave theory and applications, (1988), Oxford University Press London · Zbl 0648.73015 [4] Fowkes, N.D.; Mahony, J.J., An introduction to mathematical modelling, (1994), Wiley New York [5] Debnath, L., Nonlinear partial differential equations for scientists and engineers, (1997), Birkhäuser Boston · Zbl 0892.35001 [6] Bellomo, N.; Delitala, M.; Coscia, V., Math. models methods appl. sci., 12, 1801, (2002) [7] Jordan, P.M., Phys. lett. A, 326, 77, (2004) [8] Drazin, P.G., Solitons, (1983), Cambridge University Press Cambridge · Zbl 0517.35002 [9] Pierce, A.D., Acoustics, acoustical society of America, (1989), Woodbury NY [10] Joseph, D.D.; Preziosi, L.; Joseph, D.D.; Preziosi, L., Rev. mod. phys., Rev. mod. phys., 62, 375, (1990) [11] Özişik, M.N.; Tzou, D.Y., J. heat trans., 116, 526, (1994) [12] Keller, J.B., Proc. natl. acad. sci. USA (PNAS), 101, 1120, (2004) [13] Jordan, P.M.; Puri, A., J. appl. phys., 85, 1273, (1999) [14] Lighthill, M.J.; Whitham, G.B., Proc. R. soc. London, ser. A, 229, 317, (1955) [15] P.J. Chen, in: S. Flügge, C. Truesdell (Eds.), Handbuch der Physik, vol. VIa/3, Springer, Berlin, 1973, pp. 303-402. [16] Logan, J.D., An introduction to nonlinear partial differential equations, (1994), Wiley New York · Zbl 0834.35001 [17] Lindsay, K.A.; Straughan, B., J. appl. math. phys. (ZAMP), 27, 653, (1976) [18] Jordan, P.M.; Puri, A., Phys. lett. A, 341, 427, (2005) [19] Jordan, P.M.; Christov, C.I., J. sound vib., 281, 1207, (2005) [20] P.M. Jordan, Proc. R. Soc. London, Ser. A (in press). [21] Quintanilla, R.; Straughan, B., Proc. R. soc. London, ser. A, 460, 1169, (2004) [22] Chandrasekharaiah, D.S., Appl. mech. rev., 51, 705, (1998) [23] Daganzo, C.F., Transpn. res. B, 29, 277, (1995) [24] Lee, H.Y.; Lee, H.-W.; Kim, D., Phys. rev. lett., 81, 1130, (1998) [25] Saffman, P.G., J. fluid mech., 13, 120, (1962) [26] P.M. Jordan, A. Puri, Quart. Jl. Mech. Appl. Math. (in press).
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