Local discontinuous Galerkin method for solving Burgers and coupled Burgers equations

In the current work, we extend the local discontinuous Galerkin method to a more general application system. The Burgers and coupled Burgers equations are solved by the local discontinuous Galerkin method. Numerical experiments are given to verify the efficiency and accuracy of our method. Moreover the numerical results show that the method can approximate sharp fronts accurately with minimal oscillation.     

New exact solutions of Burgers’ type equations with conformable derivative

In this paper, the new exact solutions for some nonlinear partial differential equations are obtained within the newly established conformable derivative. We use the first integral method to establish the exact solutions for time-fractional Burgers’ equation, modified Burgers’ equation, and Burgers–Korteweg–de Vries equation. We report that this method is efficient and it can be successfully used to obtain new analytical solutions of nonlinear FDEs.     

A meshless method for the compound KdV-Burgers equation

The element-free Galerkin (EFG) method for numerically solving the compound Korteweg-de Vries-Burgers (KdVB) equation is discussed in this paper.The Galerkin weak form is used to obtain the discrete equation and the essential boundary conditions are enforced by the penalty method.The effectiveness of the EFG method of solving the compound Korteweg-de Vries-Burgers (KdVB) equation is illustrated by three numerical examples.     

多前车速度差的车辆跟驰模型的稳定性与孤波

通过线性稳定性分析,得到了多前车速度差模型的稳定性条件, 并发现通过调节多前车信息,使交通流的稳定区域明显扩大. 通过约化摄动方法 研究了该模型的非线性动力学特性:在稳定流区域,得到了描述密度波的Burgers方程;在交 通流的不稳定区域内,在临界点附近获得了描述车头间距的修正的Korteweg-de Vries (modified Korteweg-de Vries, mKdV)方程; 在亚稳态区域内,在中性稳定曲线附近获得了描述车头间距 的KdV方程. Burgers的孤波解、mKdV方程的扭结-反扭结波解及KdV方程的 孤波解描述了交通流堵塞现象.     

The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq–Burgers equation

This paper studies the coupled Burgers equation and the high-order Boussinesq–Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink (soliton) solutions and multiple-singular-kink (soliton) solutions are derived for the two equations.     

THE EXACT SOLUTIONS OF THE BURGERS EQUATION AND HIGHER-ORDER BURGERS EQUATION IN (2+1) DIMENSIONS

Some exact solutions of the Burgers equation and higher-order Burgers equation in (2+1) dimensions are obtained by using the extended homogeneous balance method. In these solutions there are solitary wave solutions, close formal solutions for the initial value problems of the Burgers equation and higher-order Burgers equation, and also infinitely many rational function solutions. All of the solutions contain some arbitrary functions that may be related to the symmetry properties of the Burgers equation and the higher-order Burgers equation in (2+1) dimensions.     

Homotopic mapping method of solitary wave solutions for generalized complex Burgers equation

A class of generalized complex Burgers equation is considered. First, a set of equations of the complex value functions are solved by using the homotopic mapping method. The approximate solution for the original generalized complex Burgers equation is obtained. This method can find the approximation of arbitrary order of precision simply and reliably.     

Painlevé Analysis and Some Solutions of(2+1)-Dimensional Generalized Burgers Equations

Burgers equation ut = 2uux + uxx describes a lot of phenomena in physics fields, and it has attracted much attention.In this paper,the Burgers equation is generalized to (2+1) dimensions.By means of the Painlev(e') analysis,the most generalized Painlev(e') integrable(2+1)-dimensional integrable Burgers systems are obtained.Some exact solutions of the generalized Burgers system are obtained via variable separation approach.     

求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法

构造一类求解三种类型偏微分方程的间断Petrov-Galerkin方法.求解的方程分别含有二阶、三阶和四阶偏导数,包括Burgers型方程、KdV型方程和双调和型方程.首先将高阶微分方程转化成为与之等价的一阶微分方程组,再将求解双曲守恒律的间断Petrov-Galerkin方法用于求解微分方程组.该方法具有四阶精度且具有间断Petrov-Galerkin方法的优点.数值实验表明该方法可以达到最优收敛阶而且可以模拟复杂波形相互作用,如孤立子的传播及相互碰撞等.     

Nonclassical Potential Symmetries and New Explicit Solutions of Burgers Equation

A new method is used to determine the nonclassical potential symmetry generators of Burgers equation.Some classes of new explicit solutions, which cannot be obtained by Lie symmetry group of Burgers equation or its integrated equation, are obtained by using these new nonclassical potential symmetry generators.     

Nonclassical Potential Symmetries and New Explicit Solutions of Burgers Equation

A new method is used to determine the nonclassical potential symmetry generators of Burgers equation.Some classes of new explicit solutions, which cannot be obtained by Lie symmetry group of Burgers equation or its integrated equation, are obtained by using these new nonclassical potential symmetry generators.     

Multi-dimensional quasi-simple waves in weakly dissipative flows

A multi-dimensional simple wave formalism is employed to formulate a multi-dimensional quasi-simple wave for a weakly dissipative fluid. This is a natural but nontrivial generalization of the so-called unidirectional quasi-simple wave. The method is more close to multi-orders analysis which is different from the standard perturbation method. Detailed solving up to the second order is presented for a 2D sound simple wave. A new 2D Burgers equation is derived for the wave phase. It essentially differs from the known 2D generalizations of the Burgers equation, e.g., from the Zabolotskaya-Khokhlov equation. The derived equation is investigated in the framework of group analysis of differential equations. Multi-parameter families of its exact solutions are constructed. Simplest solutions are chosen for analysis of their physical relevance in the initial variables.     

Viscous and inviscid regularizations in a class of evolutionary partial differential equations

We investigate solution properties of a class of evolutionary partial differential equations (PDEs) with viscous and inviscid regularization. An equation in this class of PDEs can be written as an evolution equation, involving only first-order spatial derivatives, coupled with the Helmholtz equation. A recently developed two-step iterative method (P.H. Chiu, L. Lee, T.W.H. Sheu, A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation, J. Comput. Phys. 228 (2009) 8034–8052) is employed to study this class of PDEs. The method is in principle superior for PDE’s in this class as it preserves their physical dispersive features. In particular, we focus on a Leray-type regularization (H.S. Bhat, R.C. Fetecau, A Hamiltonian regularization of the Burgers equation, J. Nonlinear Sci. 16 (2006) 615–638) of the Hopf equation proposed in alternative to the classical Burgers viscous term. We show that the regularization effects induced by the alternative model can be vastly different from those induced by Burgers viscosity depending on the smoothness of initial data in the limit of zero regularization. We validate our numerical scheme by comparison with a particle method which admits closed form solutions. Further effects of the interplay between the dispersive terms comprising the Leray-regularization are illustrated by solutions of equations in this class resulting from regularized Burgers equation by selective elimination of dispersive terms.     

Consistent Burgers equation expansion method and its applications to high- dimensional Burgers-type equations

In this paper, a novel method, named the consistent Burgers equation expansion (CBEE) method, is proposed to solve nonlinear evolution equations (NLEEs) by the celebrated Burgers equation. NLEEs are said to be CBEE solvable if they are satisfied by the CBEE method. In order to verify the effectiveness of the CBEE method, we take (2+1)-dimensional Burgers equation as an example. From the (1+1)-dimensional Burgers equation, many new explicit solutions of the (2+1)-dimensional Burgers equation are derived. The obtained results illustrate that this method can be effectively extended to other NLEEs.     

Direct perturbation method for perturbed complex Burgers equation

So far, Lou's direct perturbation method has been applied successfully to solve the nonlinear Schr?dinger equation(NLSE) hierarchy, such as the NLSE, the coupled NLSE, the critical NLSE, and the derivative NLSE. But to our knowledge, this method for other types of perturbed nonlinear evolution equations has still been lacking. In this paper, Lou's direct perturbation method is applied to the study of perturbed complex Burgers equation. By this method, we calculate not only the zero-order adiabatic solution, but also the first order modification.     

A(2+1)-dimensional nonlinear model for Rossby waves in stratified fluids and its solitary solution

In this paper,we investigate a(2+1)-dimensional nonlinear equation model for Rossby waves in stratified fluids.We derive a forced Zakharov–Kuznetsov(ZK)–Burgers equation from the quasigeostrophic potential vorticity equation with dissipation and topography under the generalized beta effect,and by utilizing temporal and spatial multiple scale transform and the perturbation expansion method.Through the analysis of this model,it is found that the generalized beta effect and basic topography can induce nonlinear waves,and slowly varying topography is an external impact factor for Rossby waves.Additionally,the conservation laws for the mass and energy of solitary waves are analyzed.Eventually,the solitary wave solutions of the forced ZK–Burgers equation are obtained by the simplest equation method as well as the new modified ansatz method.Based on the solitary wave solutions obtained,we discuss the effects of dissipation and slowly varying topography on Rossby solitary waves.     

The stabilization effect of the density difference in the modified lattice hydrodynamic model of traffic flow

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 difference leads to the stabilization of the system. The Burgers equation and mKdV equation are derived to describe the density waves in the stable and unstable regions respectively. Numerical simulations show that considering the density difference not only could stabilize traffic flow but also makes the lattice hydrodynamic model more realistic.     

Symmetry Reductions of Two-Dimensional Variable Coefficient Burgers Equation

By use of a direct method, we discuss symmetries and reductions of the two-dimensional Burgers equation with variable coefficient (VCBurgers). Five types of symmetry-reducing VCBurgers to (1+1)-dimensional partial differential equation and three types of symmetry reducing VCBurgers to ordinary differential equation are obtained.     

Solution and transcritical bifurcation of Burgers equation

Burgers equation is reduced into a first-order ordinary differential equation by using travelling wave transformation and it has typical bifurcation characteristics.We can obtain many exact solutions of the Burgers equation,discuss its transcritical bifurcation and control dynamical behaviours by extending the stable region.The transcritical bifurcation exists in the (2 + 1)-dimensional Burgers equation.     

Dynamical behaviour of stochastic dust charge fluctuations

The dynamics of stochastic dust charge fluctuations is investigated. An inhomogeneous Burgers equation with a linear forcing term is derived in which the Burgers term is proportional to the diffusivity of dust charge fluctuations. This novel equation is solved analytically. Possibility of turbulence due to stochastic charge fluctuations is highlighted.