Macroscopic Equation and Its Application for Free Ways

Abstract

Effective transportation systems lead to the efficient movement of goods and people, which significantly contribute to the quality of life in every society. In the heart of every economic and social development, there is always a transportation system. Mathematically the problem of modeling vehicle traffic flow can be solved at two main observation scales: The microscopic and the macroscopic levels. In the microscopic level, every vehicle is considered individually, and therefore, for every vehicle, we have an equation that is usually an ordinary differential equation (ODE). At a macroscopic level, we use from the dynamics models, where we have a system of partial differential equation, which involves variables such as density, speed, and flow rate of traffic stream with respect to time and space. Therefore, considering above content, this study has tried to compare solution of equation of macroscopic flow considering linear form (speed-density) and applying boundary condition that resulting to form solved is non-linear one-order partial differential equation (sharpy method) with non-linear assuming (speed and density) and consequently homographic nonlinear relation (speed-density). The recent case clearly gives more significant speeds than linear case of speed and density that can be a good scientific basis. In terms of safety for accidents and traffic signal, just as a reminder, but it is resulted of the reality that generally solutions of partial differential equations can have different forms. Therefore, the solution of partial differential equation (macroscopic flow) can have different answers and solutions so that all of these solutions apply in PDE (equation of macroscopic flow). Thus, under this condition, we can have solution of linear equation similar to greenberg or greenshield & android that are explained in logarithm and exponential function, but this article is based mostly on nonlinear solution of macroscopic equation, provided that existing nonlinear relationship between speed and density (homographic the second degree function). As mentioned above, as it gives more reliable and reasonable speeds than greenshield case, it will have more safety. This article has been provided in this field.