Submitted: Mar 2, 2013; Accepted: Apr 10, 2013; Published: Jun 10, 2013 Abstract: In this article, we present a variant approach of the generalized and improved (G'/G)–expansion method and construct some new exact traveling wave solutions with free parameters of the nonlinear evolution equations, via the Painleve integrable Burgers equation, the Boiti-Leon-Pempinelle equation and the Pochhammer...

Higher-order accurate finite-difference schemes for solving the unsteady Burgers’ equation which often arises in mathematical modeling used to solve problems in fluid dynamics are presented. The unsteady Burgers’ equation belongs to a few nonlinear partial differential equations which has an exact solution, and it allows one to compare the numerical solution with the exact one, and the properti...

A generalization of the Hopf–Cole transformation and its relation to the Burgers equation of integer order and the diffusion equation with quadratic nonlinearity are discussed. The explicit form of a particular analytical solution is presented. The existence of the travelling wave solution and the interaction of nonlocal perturbation are considered. The nonlocal generalizations of the one-dimen...

We consider the problem of stabilization of unstable “shock-like” equilibrium profiles of the viscous Burgers’ equation with actuation at the boundaries. These equilibria are not stabilizable (even locally) using the standard “radiation boundary condition.” Using a nonlinear spatially-scaled transformation (that employs three ingredients, of which one is the Hopf-Cole nonlinear integral transfo...

This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers’ equation. Firstly, is derived deal system ordinary differential equations (ODEs) arising from semi-discretization initial-boundary value partial (PDEs). Further, method this considered as special case and fourth-order difference resulting in combined nu...

The purpose of this paper is to investigate the relation between the moments and the asymptotic behavior of solutions to the Burgers equation. The Burgers equation is a special nonlinear problem that turns into a linear one after the Cole-Hopf transformation. Our asymptotic analysis depends on this transformation. In this paper an asymptotic approximate solution is constructed, which is given b...

In this paper we construct a stochastic particle method for the Burgers equation with a monotone initial condition; we prove that the convergence rate is O(1= p N + p t) for the L 1 (IR)-norm of the error. To obtain that result, we link the PDE and the algorithm to a system of weakly interacting stochastic particles; the diiculty of the analysis comes from the discontinuity of the interaction k...

In this paper we study the dynamics of a Burgers’ type equation (1.1). First, we use a new method called attractor bifurcation introduced by Ma and Wang in [4, 6] to study the bifurcation of Burgers’ type equation out of the trivial solution. For Dirichlet boundary condition, we get pitchfork attractor bifurcation as the parameter λ crosses the first eigenvalue. For periodic boundary condition,...

In this paper, we provide a blow-up mechanism to the modified Camassa-Holm equation with varying linear dispersion. We first consider the case when linear dispersion is absent and derive a finite-time blow-up result with an initial data having a region of mild oscillation. A key feature of the analysis is the development of the Burgers-type inequalities with focusing property on characteristics...