In this paper, the modified Adomian s decomposition method for calculating a numerical solution of the one-dimensional quasi-linear, the Burgers equation, is presented. Time discretization has been used in decomposition, without using any transformation in the Burgers equation such as Hopf–Cole transformation. The numerical results obtained by this way for various values of viscosity have been ...

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By the Cole-Hopf transformation, with any linear evolution equation in 1 + 1 dimensions a generalized Burgers equation is associated. We describe local conservation laws of these equations. It turns out that any generalized Burgers equation has only one conservation law, while a linear evolution equation with constant coefficients has an infinite number of (x, t)independent conservation laws if...

In this paper, we extend the classical Lie symmetry analysis from partial differential equations to integro-differential equations with functional derivatives. We continue the work of Oberlack and Wacławczyk (2006, Arch. Mech. 58, 597), (2013, J. Math. Phys. 54, 072901), where the extended Lie symmetry analysis is performed in the Fourier space. Here, we introduce a method to perform the extend...

We study Lagrangian trajectories and scalar transport statistics in decaying Burgers turbulence. We choose velocity fields solutions of the inviscid Burgers equation whose probability distributions are specified by Kida’s statistics. They are time-correlated, not time-reversal invariant and not Gaussian. We discuss in some details the effect of shocks on trajectories and transport equations. We...

Consider the viscous Burgers equation on a bounded interval with inhomogeneous Dirichlet boundary conditions. Following the variational framework introduced in [4], we analyze a Lyapunov functional for such equation which gives the large deviations asymptotics of a stochastic interacting particles model associated to the Burgers equation. We discuss the asymptotic behavior of this energy functi...

We study the effect of viscosity on the large time behavior of the viscous Burgers equation by using a transformed version of Burgers (in self-similar variables) that captures efficiently the mechanism of transition to the asymptotic states and allows us to estimate the time of evolution from an N-wave to the final stage of a diffusion wave. Then we construct certain special solutions of diffus...

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...

We report on the first direct nanoscale imaging of elementary edge dislocations in a thermotropic smectic-C* liquid crystal with the Burgers vector equal to one smectic layer spacing d. We find two different types of dislocation profiles. In the dislocation of type A, the layers deformations lack mirror symmetry with respect to the plane perpendicular to the Burgers vector; the dislocation core...