We propose a difference-differential equation that reflects interactions between innovation and imitation processes to describe the evolution of the distribution curve of firms by efficiency levels. An explicit solution of this equation is obtained for arbitrary finite initial conditions. It is...

A continuum version of the car-following full velocity difference model (FVDM) is developed using a series expansion of the headway in terms of the density. This continuum model obeys the equivalent stability criterion as its discrete counterpart, and the Burgers and Korteweg-de Vries equations...

A modified lattice hydrodynamic model of traffic flow is proposed by introducing the density difference between the leading and the following lattice. The stability condition of the modified model is obtained through the linear stability analysis. The results show that considering the density...

We study the formation of shockwaves from an initial condition of the pulse form in supercritical flow of traffic by using the optimal velocity model. The jam with the pulse form propagates with changing the initial form. The wave velocity is derived numerically and analytically. The dependence...

We study the dynamics of the Burgers equation on the unit interval driven by affine linear noise. Mild solutions of the Burgers stochastic partial differential equation generate a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques,...

A nonstandard finite difference scheme is constructed for the Burgers partial differential equation having no diffusion and a nonlinear logistic reaction term. This scheme preserves the positivity and boundedness properties of the original differential equation and includes the a priori...

We construct a nonstandard finite difference (NSFD) scheme for a Burgers type partial differential equation (PDE) for which the diffusion coefficient has a linear dependence on the dependent variable. After a study of this PDE's traveling-wave solutions, we examine the corresponding properties...

We consider charge transport in a pore where the dielectric constant inside the pore is much greater than that in the surrounding material, so that the flux of the electric fields due to the charges is almost entirely confined to the pore. We develop exact solutions for the one component case...

With the aim of gaining insight into the notoriously difficult problem of energy and vorticity cascades in high dimensional incompressible flows, we take a simpler and very well understood low dimensional analog and approach it from a new perspective, using the Fourier transform. Specifically,...

We show that the critical mass Mc=8π of bacterial populations in two dimensions in the chemotactic problem is the counterpart of the critical temperature Tc=GMm/4kB of self-gravitating Brownian particles in two-dimensional gravity. We obtain these critical values by using the Virial theorem or...