The Burgers’ equation is one of the typical nonlinear evolutionary partial differential equations. In this paper, a mesh-free method is proposed to solve the Burgers’ equation using the finite difference and collocation methods. With the temporal discretization of the equation using C-N scheme, the solution is approximated spatially by Radial Basis Function (RBF). The numerical results of two d...

The instanton solution for the forced Burgers equation is found. This solution describes the exponential tail of the probability distribution function of velocity differences in the region where shock waves are absent. The results agree with the one found recently by Polyakov, who used the operator product conjecture. If this conjecture is true, then our WKB asymptotics of the Wyld functional i...

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...

Earlier results for distributed and boundary controls of the viscous Burgers' equation were established by Burns et al. and Byrnes et al.. In their results there are technical restrictions on the sizes of the initial data. In this paper we relax these restrictions, as well as treat the Burgers' equation with other nonlinear boundary conditions.

The main objective of this article is to analyse the dynamics of Burgers equation on the unit interval, driven by multiplicative white noise. It is shown that the solution field of the stochastic Burgers equation generates a smooth perfect and locally compacting cocycle on the energy space L2([0, 1],R). Using multiplicative ergodic theory techniques, we compute the discrete non-random Lyapunov ...

The Burgers’ equation is an important and basic nonlinear partial differential equation in fluid dynamics, and has been used as a model equation in other fields, such as modeling of shock waves, gas dynamics, turbulence, and large bubble structures consisting of clusters of galaxies in space. Many researchers have proposed various numerical methods for solving the Burgers’ equation, such as the...

In this article, we prove that the Kolmogorov operator associated to the Burgers equation driven by a space-time white noise is m-dissipative. This implies several properties on the Kolmogorov equation. This result is obtained thanks to the introduction of a modified Kolmogorov operator. New a priori estimates on the solutions of the Burgers equation and on the invariant measure are obtained. T...