Non-Linearity Flux of Fractional Transport Density Equation in Traffic Flow with Solutions

In the present paper, we derive and solve the space-fractional traffic flow model which is considered as a generalization of the transport density equation. Based on the fundamental physical principles on finite-length highway where the number of vehicles is conserved, without entrances or exits, we construct a fractional continuity equation. As a limitation of the classical calculus, the continuity equation is constructed based on truncating after the first order of Taylor expansion, which means that the change in the number of vehicles is linear over the finite-length highway. However, in fractional calculus, we prove that nonlinear flow is a result of truncating the fractional Taylor polynomial after the second term with zero error. Therefore, the new fractional traffic flow model is free from being linear, and the space now is described by the fractional powers of coordinates, provided with a single variable measure. Further, some exact solutions of the fractional model are generated by the method of characteristics. Remarkably, these solutions have significant physical implications to help to make the proper decisions for constructing traffic signals in a smart city.