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 ...

We construct new integrable coupled systems of N = 1 supersymmetric equations and present integrable fermionic extensions of the Burgers and Boussinesq equations. Existence of infinitely many higher symmetries is demonstrated by the presence of recursion operators. Various algebraic methods are applied to the analysis of symmetries, conservation laws, recursion operators, and Hamiltonian struct...

In this paper we establish rigorously that the family of Burgers vortices of the three-dimensional Navier-Stokes equation is stable for small Reynolds numbers. More precisely, we prove that any solution whose initial condition is a small perturbation of a Burgers vortex will converge toward another Burgers vortex as time goes to infinity, and we give an explicit formula for computing the change...

In this paper we establish rigorously that the family of Burgers vortices of the three-dimensional Navier-Stokes equation is stable for small Reynolds numbers. More precisely, we prove that any solution whose initial condition is a small perturbation of a Burgers vortex will converge toward another Burgers vortex as time goes to infinity, and we give an explicit formula for computing the change...

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 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 recent systematic study by Janocha et al. [1] to determine all possible Lie-point symmetries for the functional Hopf–Burgers equation is re-examined. From a more consistent theoretical framework, however, some of the proposed symmetry transformations of the considered Hopf–Burgers equation are in fact rejected. Three out of eight proposed symmetry transformations are invalidated, while two ...