The complex -symmetric nonlinear wave models have drawn much attention in recent years since the complex -symmetric extensions of the Korteweg-de Vries (KdV) equation were presented in 2007. In this review, we focus on the study of the complex -symmetric nonlinear Schrödinger equation and Burgers equation. First of all, we briefly introduce the basic property of complex symmetry. We then report...

The Burgers' model of compressible fluid dynamics in one dimension is extended to include the effects of pressure back-reaction. The system consists of two coupled equations: Burgers' equation with a pressure gradient (essentially the one-dimensional Navier-Stokes equation) and an advection-diffusion equation for the pressure field. It presents a minimal model of both adiabatic gas dynamics and...

In this paper, analytic solutions of two-dimensional coupled Burgers' equations are obtained by the Homotopy analysis and the Homotopy Pad$acute{e}$ methods. The obtained approximation by using Homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by $[m,m]$ Homotopy Pad$acute{e...

Non-local transformation, which connects the 3-dimensional Burgers-type equation with a linear heat equation, is constructed. Via this transformation, nonlinear superposition formulae for solutions are obtained and the conditional non-local symmetry of this equation is studied. The multidimensional generalization of the Burgers equation L1(u) = u0 − u|∇u| − u = 0, (1) is called further the Burg...

In this work we carry out a complete group classification of Burgers’ equations. 2000 AMS Mathematics Classification numbers: 22E60, 35K55, 35Q53, 58J70

We describe new classes of nonlinear Galilean–invariant equations of Burgers and Korteweg–de Vries type and study symmetry properties of these equations.

In the current paper the e ciency of the sparse-grid combination technique applied to time-dependent advectiondi usion problems is investigated. For the time integration we employ a third-order Rosenbrock scheme implemented with adaptive step-size control and approximate matrix factorization. Two model problems are considered, a scalar 2D linear, constant-coe cient problem and a system of 2D no...