Application Of Lie Group Reduction Method To The Full Velocity Difference Model And Its Time Delay Model

In recent years, urban traffic congestion has become a major problem affecting on public travel and economic sustainable development. The study of traffic flow model becomes a new hot topic again. The car-following models are differential-difference models for describing the flow law of the vehicles. To obtain the solitary wave and kink solutions of these differential-difference models has very important significance both in theory and application for analyzing traffic congestion.Lie group reduction method is an effective tool for solving the group invariant, kink and solitary wave solutions of partial differential equations. In this paper, we shall apply the Lie group reduction method to the full velocity difference model(FVDM)and its time delay model and obtain some new kink and solitary wave solutions for the first time.At first, after changing the FVDM to a continuous forth-order nonlinear partial differential equation, we perform the Lie symmetry analysis of the equation, and obtain its infinitesimal generators, Lie algebra commutator table, one-parameter symmetry groups and the corresponding similarity variables. We derive a reduction equation that is a third-order nonlinear ordinary differential equation with the similarity variables. Furthermore, we obtain the invariant, kink and solitary wave solutions of the FVDM by solving the reduction equation with tanh and sech function expansion methods respectively. From the images analysis of the kink and solitary wave solutions, we know that the headway decreases as the maximum velocity increases when the flow quantity of vehicles is certain and that the headway decreases as the flow quantity of vehicles increases when the maximum velocity is certain. We also show that the sensitivity coefficient has positive relationship with the headway.Then, after changing the time delay model of the FVDM to a continuous forth-order nonlinear partial differential equation, we perform the Lie symmetry analysis of the equation, and obtain its infinitesimal generators, Lie algebra commutator table, one-parameter symmetry groups and the corresponding similarity variables.We derive a reduction equation that is a third-order nonlinear ordinary differential equations with the similarity variables. Furthermore, we obtain the invariant and kink solutions of it by solving the reduction equation with tanh function expansion method. The images analysis of the kink solutions shows that the headway decreases as the time delay increases,when the flow quantity of vehicles is certain.