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...

The dynamics of edge dislocations with parallel Burgers vectors, moving in the same slip plane, is mapped onto Dyson's model of a two-dimensional Coulomb gas confined in one dimension. We show that the tail distribution of the velocity of dislocations is power law in form, as a consequence of the pair interaction of nearest neighbors in one dimension. In two dimensions, we show the presence of ...

Direct algebraic method of obtaining exact solutions to nonlinear PDE’s is applied to certain set of nonlinear nonlocal evolutionary equations, including nonlinear telegraph equation, hyperbolic generalization of Burgers equation and some spatially nonlocal hydrodynamic-type model. Special attention is paid to the construction of the kink-like and soliton-like solutions.

The Moyal *-deformed noncommutative version of Burgers' equation is considered. Using the *-analog of the Cole-Hopf transformation, the linearization of the model in terms of the linear heat equation is found. Noncommutative q-deformations of shock soliton solutions and their interaction are described

This paper discusses the application of orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence name collocation. The used solve various cases Burgers’ equations, including modified equation. KdV–Burgers’ equation considered as a tes...

We consider a time discrete kinetic scheme (known as “transport collapse method”) for the inviscid Burgers equation ∂tu+ ∂x u 2 = 0. We prove the convergence of the scheme by using averaging lemmas without bounded variation estimate. Then, the extension of this result to the kinetic model of Brenier and Corrias is discussed.

The dynamics of randomly forced Burgers and Euler-Lagrange equations in S 1 × R d−1 in the case when there is only one one-sided minimizer in a compact subset of S 1 × R d−1 is studied. The existence of random invariant periodic minimizer orbits and periodicity of the stationary solution of the stochastic Burgers equations are obtained.

We generelize the Polyakov’s approach for Burgers turbulence in higher dimensions. In this respect, we write the operator product expansion and find the exact two–point functions of the Burgers equation in two and three–dimensions. We show that the angular dependence of the correlation functions satisfy the same equation, which is found in the instanton approach. PACS numbers 47.27.AK, 47.27.Jv

In this paper, numerical solution for the generalized Rosenau-Burgers equation is considered and Crank-Nicolson finite difference scheme is proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence and priori error estimate of the scheme are proved. Numerical results demonstrate that the scheme is efficient and reliable. Keywords—Generalized Rosenau-B...

We have studied the phase dynamics in an oscillatory chemical system (Belousov-Zhabotinsky reaction). The phase distribution and isophase lines in the 2D phase space were reconstructed. The dynamics proved to be well described in terms of the Burgers equation. The phase dependence of the coefficients of the equation was taken into account to estimate the region of the applicability of the Burge...