Vol. 6, No. 1 (2017): Regular Issue

In this paper, the classical symmetries and the potential symmetries of KdVBurgers equation are calculated based on differential characteristic set algorithm, and the corresponding Lie transformation groups and invariant solutions of the potential symmetry are derived. Moreover a series of new exact solutions for KdV-Burgers equation are obtained by acting Lie transformation groups on the invariant solutions. It is important that these solutions can not be obtained from the classical symmetries of KdV-Burgers equation.

One-dimensional gas filtration was described nonlinear parabolic equation as the conservation law. The potential of the conservation law satisfies a equation as the conservation law. The introduce of the second potential gives a system of equations which admits 6-dimensional Lie algebra. This extends group properties of initial model. With the help of optimal system of subalgebras are classified all invariant and partial invariant solutions which are reduced to invariant solutions of the initial model. Some time it is possible to find integral of invariant submodel.

In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractivemultivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We study a new phase space for impulsive system with infinite delay.

We consider an approach which allows to describe uniformly in time the process of collision of solitary waves. Next we apply it to the KdV-type equation with nonlinearity u4 for three interacting waves assuming that all wave trajectories intersect at the same point. The constructed asymptotic solution satisfies the equation in a weak sense and it can be treated as a classical asymptotics in the sense that it satisfies some conservation and balance laws associated with the gKdV-4 equation. Results of numerical simulation confirm the theoretical conclusion about the elastic type of the wave interaction.

Fractional differential equations with random impulses arise in modeling real world phenomena where the state changes instantaneously at uncertain moments. Using queuing theory and the usual distribution for waiting time, we study the case of exponentially distributed random variables between two consecutive moments of impulses. The p-moment exponential stability of solutions is defined and studied when the waiting time between two consecutive impulses is exponentially distributed. The argument is based on Lyapunov functions. We discuss both continuous and differentiable Lyapunov functions and Caputo fractional Dini derivatives and Caputo derivatives are applied. Some examples are given to illustrate our results.

Using the Chiellini condition for integrability we derive explicit solutions for a generalized system of Riccati equations x+αx2n+1x+x4n+3 = 0 by reduction to the first-order Abel equation assuming the parameter α ≥ 2 2(n+1). The technique, which was proposed by Harko et al, involves use of an auxiliary system of first-order differential equations sharing a common solution with the Abel equation. In the process analytical proofs of some of the conjectures made earlier on the basis of numerical investigations in [1] is provided.

We establish the existence of stationary solutions for certain systems of reaction-diffusion equations with superdiffusion. The corresponding elliptic problem involves the operators with or without Fredholm property. The fixed point technique in appropriate H2 spaces of vector functions is employed.

One of the challenges of autonomous ground vehicles (AGVs) is to interact with human driven vehicles in the traffic. This paper develops defensive driving strategies for AGVs to avoid problematic vehicles in the mixed traffic. A multi-objective optimization algorithm for local trajectory planning is proposed. The dynamic predictive control is used to derive optimal trajectories in a rolling horizon. The intelligent driver model and lanechanging rules are employed to predict the movement of the vehicles. Multiple performance objectives are optimized simultaneously, including traffic safety, transportation efficiency, driving comfort and path consistency. The multi-objective optimization problem is solved with the cell mapping method. Different and relatively simple scenarios are created to test the effectiveness of the defensive driving strategies. Extensive experimental simulations show that the proposed defensive driving strategy is promising and may provide a new tool for designing the intelligent navigation system that helps autonomous vehicles to drive safely in the mixed traffic.