The purpose of this paper is to investigate the relation between the moments and the asymptotic behavior of solutions to the Burgers equation. The Burgers equation is a special nonlinear problem that turns into a linear one after the Cole-Hopf transformation. Our asymptotic analysis depends on this transformation. In this paper an asymptotic approximate solution is constructed, which is given b...

In this work we generate the numerical solutions of Burgers’ equation by applying the Crank-Nicholson method and different schemes for solving nonlinear systems, instead of using Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. The method is analyzed on two test problems in order to check its efficiency on different kinds of initial conditions. Numerical solut...

This paper examines the properties of a regularization of the Burgers equation in one and multiple dimensions using a filtered convective velocity, which we have dubbed as the convectively filtered Burgers (CFB) equation. A physical motivation behind the filtering technique is presented. An existence and uniqueness theorem for multiple dimensions and a general class of filters is proven. Multip...

In a famous calculation of the “large-distance and longtime properties of a randomly stirred fluid,” Forster, Nelson, and Stephen (FNS) analyzed the consequences of various forcing scenarios for the Navier–Stokes equation and, to some extent, Burgers equation. They considered both a conservative forcing (Model A) and a nonconservative one (Model B), and predicted nontrivial properties for the l...

New solutions of some important partial differential equations are obtained using the first integral method. The efficiency of the method is demonstrated by applying it for the linear Klein-Gordan equation, MKdV equation, Burgers’ equation in two and three dimensions.

Approximation theorems, analogous to known results for linear elliptic equations, are obtained for solutions of the heat equation. Via the Cole-Hopf transformation, this gives rise to approximation theorems for a nonlinear parabolic equation, Burgers’ equation.

We propose the set of coupled ordinary differential equations dnj/dt = nj−1 − nj as a discrete analogue of the classic Burgers equation. We focus on traveling waves and triangular waves, and find that these special solutions of the discrete system capture major features of their continuous counterpart. In particular, the propagation velocity of a traveling wave and the shape of a triangular wav...

An elementary yet remarkable similarity between the Cole-Hopf transformation relating the Burgers and heat equation and Miura's transformation connecting the KdV and mKdV equations is studied in detail. 1. Introduction Our aim in this note is to display the close similarity between the well-known Cole{Hopf transformation relating the Burgers and the heat equation, and the celebrated Miura trans...

We make use of the method of modulus of continuity [7] and Fourier localization technique [1] to prove the global well-posedness of the critical Burgers equation ∂tu + u∂xu + Λu = 0 in critical Besov spaces Ḃ 1 p p,1(R) with p ∈ [1,∞), where Λ = √ −△. 2000 Mathematics Subject Classification: 35K55, 35Q53

We consider Burgers equation forced by a brownian in space and white noise in time process ∂tu+ 1 2 ∂x(u) = f(x, t), with E(f(x, t)f(y, s)) = 1 2 (|x|+ |y| − |x− y|)δ(t− s) and we show that there are Levy processes solutions, for which we give the evolution equation of the characteristic exponent. In particular we give the explicit solution in the case u0(x) = 0.