# American Institute of Mathematical Sciences

February  2017, 14(1): 31-43. doi: 10.3934/mbe.2017003

## On the initial value problem for a class of discrete velocity models

 University of Ferrara, Department of Mathematics and Computer Science, Via Machiavelli 35,44121 Ferrara, Italy

Received  October 22, 2015 Accepted  January 12, 2016 Published  October 2016

In this paper we investigate the initial value problem for a class of hyperbolic systems relating the mathematical modeling of a class of complex phenomena, with emphasis on vehicular traffic flow. Existence and uniqueness for large times of solutions, a basic requisite both for models building and for their numerical implementation, are obtained under weak hypotheses on the terms modeling the interaction among agents. The results are then compared with the existing literature on the subject.

Citation: Davide Bellandi. On the initial value problem for a class of discrete velocity models. Mathematical Biosciences & Engineering, 2017, 14 (1) : 31-43. doi: 10.3934/mbe.2017003
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Initial data corresponding to $\rho_c=5$. The class $\overline{f}_1$ of velocity $v_1=0$ is on the right in red color, the class $\overline{f}_2$ of velocity $v_2=1$ is represented by the bimodal distribution on the left in blue color
Final distribution functions corresponding to $\Delta=0.5, 0.4,0.35,0.25$, and initial data as in Figure 1
Maximum $\rho_{t^*}$ of the density reached at time $t^*$ as function of the size $\Delta$ of the interaction domain $D_x$. Observe that in any case $\rho_{t^*}$ is greater that $\rho_c$
 $\Delta$ $t^*$ $\rho_{t^*}$ 1.05 4.5425 5.1890 1.00 4.5885 5.2214 0.95 4.6365 5.2660 0.90 4.6855 5.3269 0.85 4.7365 5.4092 0.80 4.7925 5.5192 0.75 4.8555 5.6652 0.70 4.9255 5.8575 0.65 5.0025 6.1090 0.60 5.4595 6.4658 0.55 5.4985 6.9726 0.50 5.5406 7.5731 0.45 5.5786 8.3056 0.40 5.6106 9.2196 0.35 5.6346 10.3606 0.30 5.6536 11.7122 0.25 5.6706 13.0508 0.20 5.6846 13.7962 0.15 5.6926 13.3582
 $\Delta$ $t^*$ $\rho_{t^*}$ 1.05 4.5425 5.1890 1.00 4.5885 5.2214 0.95 4.6365 5.2660 0.90 4.6855 5.3269 0.85 4.7365 5.4092 0.80 4.7925 5.5192 0.75 4.8555 5.6652 0.70 4.9255 5.8575 0.65 5.0025 6.1090 0.60 5.4595 6.4658 0.55 5.4985 6.9726 0.50 5.5406 7.5731 0.45 5.5786 8.3056 0.40 5.6106 9.2196 0.35 5.6346 10.3606 0.30 5.6536 11.7122 0.25 5.6706 13.0508 0.20 5.6846 13.7962 0.15 5.6926 13.3582
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