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## ‣ Soluções analíticas da equação de Burgers aplicada à formação de estruturas no Universo

Fonte: Universidade de Brasília
Publicador: Universidade de Brasília

Tipo: Dissertação

Português

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Dissertação (mestrado)-Universidade de Brasília, Instituto de Física, Brasília, 2008.; A equacao de Burgers atualmente tem sido aplicada a varias areas do conhecimento cientifico, principalmente no estudo de formacao de estruturas no Universo. Sua relevancia vem aumentando a cada dia, devido `a riqueza de dados observacionais que atualmente existe na literatura moderna. Sua forma mais geral e conhecida como equacao generalizada de Burgers com ruido e foi proposta por Ribeiro e Peixoto de Faria (2005). Conhecer suas solucoes exatas e escritas de forma claraedemuitointeresseastrofisico. Comesseintuitoapresentamos, nestetrabalho, solucoes invariantes sob simetrias de Lie da equacao generalizada de Burgers sem o termo estocastico, obtidas a partir do pacote de analises de simetrias de equacoes diferenciais (SADE) escrito em MAPLE, desenvolvido no IF-UnB. Posteriormente, simulamos uma distribuicao de velocidades a partir de algumas solucoes invariantes escolhidasdentreas220obtidas, ecomparamoscomumadistribuicaodevelocidades peculiares observacionais.
_________________________________________________________________________________ ABSTRACT; The Burgers equation has been applied to several fields of scientific knowledge, and particularly to the study of formation of structures in Universe. His relevancestillincreases...

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## ‣ Reduction operators of Burgers equation

Fonte: Academic Press
Publicador: Academic Press

Tipo: Artigo de Revista Científica

Publicado em 01/02/2013
Português

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The solution of the problem on reduction operators and nonclassical reductions of the Burgers equation is systematically treated and completed. A new proof of the theorem on the special “no-go” case of regular reduction operators is presented, and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case. All possible nonclassical reductions of the Burgers equation to single ordinary differential equations are exhaustively described. Any Lie reduction of the Burgers equation proves to be equivalent via the Hopf–Cole transformation to a parameterized family of Lie reductions of the linear heat equation.

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## ‣ Some Travelling Wave Solutions of KdV-Burgers Equation

Fonte: Hikari Ltd
Publicador: Hikari Ltd

Tipo: Artigo de Revista Científica

Português

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68.239404%

In this paper we study the extended Tanh method to obtain some
exact solutions of KdV-Burgers equation. The principle of the Tanh
method has been explained and then apply to the nonlinear KdV-
Burgers evolution equation. A finnite power series in tanh is considered
as an ansatz and the symbolic computational system is used to obtain
solution of that nonlinear evolution equation. The obtained solutions
are all travelling wave solutions.

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## ‣ Low-dimensional modelling of a generalized Burgers equation

Fonte: Research India Publications
Publicador: Research India Publications

Tipo: Artigo de Revista Científica

Publicado em //2007
Português

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Burgers equation is one of the simplest nonlinear partial differential equations—it combines the basic processes of diffusion and nonlinear steepening. In some applications it is appropriate for the diffusion coefficient to be a time-dependent function. Using a Wayne's transformation and centre manifold theory, we derive lmode and 2-mode centre manifold models of the generalised Burgers equations for bounded smooth time dependent coefficients. These modellings give some interesting extensions to existing results such as the similarity solutions using the similarity method.; http://arxiv.org/abs/math-ph/0307064; Zhenquan Li and A.J. Roberts

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## ‣ Simetrias de Lie da equação de Burgers generalizada; Lie point symmetries of generalized Burgers¿ equation

Fonte: Biblioteca Digital da Unicamp
Publicador: Biblioteca Digital da Unicamp

Tipo: Dissertação de Mestrado
Formato: application/pdf

Publicado em 03/11/2011
Português

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#Burgers#Equação de#Equação de calor#Lie#Simetrias de#Hopf-Cole#Transformação de#Burgers' equation#Heat equation#Lie point symmetry#Hopf-Cole transformation

Neste trabalho, é estudada uma generalização da equação de Burgers do ponto de vista da teoria de simetrias de Lie; In this work, a generalization of Burgers equation is studied from the point of view of Lie point symmetry theory

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## ‣ A stochastic Burgers equation from a class of microscopic interactions

Fonte: IMS
Publicador: IMS

Tipo: Artigo de Revista Científica

Publicado em //2015
Português

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#KPZ equation#Burgers#Weakly asymmetric#Zero-range#Kinetically constrained#Equilibrium fluctuations#Speed-change#Fluctuations

We consider a class of nearest-neighbor weakly asymmetric mass conservative particle systems evolving on $\mathbb{Z}$, which includes zero-range and types of exclusion processes, starting from a perturbation of a stationary state. When the weak asymmetry is of order $O(n^\gamma)$ for $1/2<\gamma\leq 1$, we show that the scaling limit of the fluctuation field, as seen across process characteristics, is a generalized Ornstein-Uhlenbeck process. However, at the critical weak asymmetry when $\gamma = 1/2$, we show that all limit points solve a martingale problem which may be interpreted in terms of a stochastic Burgers equation derived from taking the gradient of the KPZ equation. The proofs make use of a sharp `Boltzmann-Gibbs' estimate which improves on earlier bounds.; Fundação para a Ciência e a Tecnologia (FCT)

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## ‣ New conservation laws for inviscid Burgers equation

Fonte: Sociedade Brasileira de Matemática Aplicada e Computacional
Publicador: Sociedade Brasileira de Matemática Aplicada e Computacional

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 01/01/2012
Português

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#Lie point symmetry#Ibragimov's Theorem#conservation laws#inviscid Burgers equation#nonlinear self-adjoint equations#weak self-adjoint equations

In this paper it is shown that the inviscid Burgers equation is nonlinearly self-adjoint. Then, from Ibragimov's theorem on conservation laws, local conserved quantities are obtained. Mathematical subject classification: Primary: 76M60; Secondary: 58J70.

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## ‣ Spontaneous Stochasticity and Anomalous Dissipation for Burgers Equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We develop a Lagrangian approach to conservation-law anomalies in weak
solutions of inviscid Burgers equation, motivated by previous work on the
Kraichnan model of turbulent scalar advection. We show that the entropy
solutions of Burgers possess Markov stochastic processes of (generalized)
Lagrangian trajectories backward in time for which the Burgers velocity is a
backward martingale. This property is shown to guarantee dissipativity of
conservation-law anomalies for general convex functions of the velocity. The
backward stochastic Burgers flows with these properties are not unique,
however. We construct infinitely many such stochastic flows, both by a
geometric construction and by the zero-noise limit of the Constantin-Iyer
stochastic representation of viscous Burgers solutions. The latter proof yields
the spontaneous stochasticity of Lagrangian trajectories backward in time for
Burgers, at unit Prandtl number. It is conjectured that existence of a backward
stochastic flow with the velocity as martingale is an admissibility condition
which selects the unique entropy solution for Burgers. We also study linear
transport of passive densities and scalars by inviscid Burgers flows. We show
that shock solutions of Burgers exhibit spontaneous stochasticity backward in
time for all Prandtl numbers...

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## ‣ The relativistic Burgers equation on a de Sitter spacetime. Derivation and finite volume approximation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/05/2015
Português

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#Mathematics - Analysis of PDEs#Mathematical Physics#Mathematics - Numerical Analysis#35L02, 35L65, 65M08

The inviscid Burgers equation is one of the simplest nonlinear hyperbolic
conservation law which provides a variety examples for many topics in nonlinear
partial differential equations such as wave propagation, shocks and
perturbation, and it can easily be derived by the Euler equations of
compressible fluids by imposing zero pressure in the given system. Recently,
several versions of the relativistic Burgers equations have been derived on
different geometries such as Minkowski (flat), Schwarzshild and FLRW spacetimes
by LeFloch and his collaborators. In this paper, we consider a family member of
the FLRW spacetime so-called the de Sitter background, introduce some important
features of this spacetime geometry with its metric and derive the relativistic
Burgers equation on it. The Euler system of equations on the de Sitter
spacetime can be found by a known process by using the Christoffel symbols and
tensors for perfect fluids. We applied the usual techniques used for the
Schwarzshild and FLRW spacetimes in order to derive the relativistic Burgers
equation from the vanishing pressure Euler system on the de Sitter background.
We observed that the model admits static solutions. In the final part, we
examined several numerical illustrations of the given model through a finite
volume approximation based on the paper by LeFloch et al. The effect of the
cosmological constant is also numerically analysed in this part. Furthermore...

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## ‣ A Regularization of Burgers Equation using a Filtered Convective Velocity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/06/2008
Português

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This paper examines the properties of a regularization of Burgers equation in
one and multiple dimensions using a filtered convective velocity, which we have
dubbed as 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.
Multiple invariants of motion are found for the CFB equation and are compared
with those found in viscous and inviscid Burgers equation. Traveling wave
solutions are found for a general class of filters and are shown to converge to
weak solutions of inviscid Burgers equation with the correct wave speed.
Accurate numerical simulations are conducted in 1D and 2D cases where the shock
behavior, shock thickness, and kinetic energy decay are examined. Energy
spectrum are also examined and are shown to be related to the smoothness of the
solutions.

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## ‣ Simulation of the Burgers equation by NMR quantum information processing

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/10/2004
Português

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We report on the implementation of Burgers equation as a type-II quantum
computation on an NMR quantum information processor. Since the flow field
evolving under the Burgers equation develops sharp features over time, this is
a better test of liquid state NMR implementations of type-II quantum computers
than the previous examples using the diffusion equation. In particular, we show
that Fourier approximations used in the encoding step are not the dominant
error. Small systematic errors in the collision operator accumulate and swamp
all other errors. We propose, and demonstrate, that the accumulation of this
error can be avoided to a large extent by replacing the single collision
operator with a set of operators with random errors and similar fidelities.
Experiments have been implemented on 16 two-qubit sites for eight successive
time steps for the Burgers equation.; Comment: 5 pages, 3 figures

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## ‣ Suppression of unbounded gradients in a SDE associated with the Burgers equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We consider the Langevin equation describing a stochastically perturbed by
uniform noise non-viscous Burgers fluid and introduce a deterministic function
that corresponds to the mean of the velocity when we keep the value of position
fixed. We study interrelations between this function and the solution of the
non-perturbed Burgers equation. Especially we are interested in the property of
the solution of the latter equation to develop unbounded gradients within a
finite time. We study the question how the initial distribution of particles
for the Langevin equation influences this blowup phenomenon. It is shown that
for a wide class of initial data and initial distributions of particles the
unbounded gradients are eliminated. The case of a linear initial velocity is
particular. We show that if the initial distribution of particles is uniform,
then the mean of the velocity for a given position coincides with the solution
of the Burgers equation and in particular does not depend on the constant
variance of the stochastic perturbation. Further, for a one space space
variable we get the following result: if the decay rate of the power-behaved
initial particles distribution at infinity is greater or equal $|x|^{-2},$ then
the blowup is suppressed...

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## ‣ A variational approach to the stationary solutions of Burgers equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/08/2010
Português

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Consider the viscous Burgers equation on a bounded interval with
inhomogeneous Dirichlet boundary conditions. Following the variational
framework introduced by Bertini-De Sole-Gabrielli-Jona-Lasinio-Landim C, 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
functional, whose minimizer is given by the unique stationary solution, as the
length of the interval diverges. We focus on boundary data corresponding to a
standing wave solution to the Burgers equation in the whole line. In this case,
the limiting functional has in fact a one-parameter family of minimizers and we
analyze the so-called development by Gamma-convergence; this amounts to compute
the sharp asymptotic cost corresponding to a given shift of the stationary
solution.

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## ‣ Aspects of the stochastic Burgers equation and their connection with turbulence

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/05/2000
Português

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We present results for the 1 dimensional stochastically forced Burgers
equation when the spatial range of the forcing varies. As the range of forcing
moves from small scales to large scales, the system goes from a chaotic,
structureless state to a structured state dominated by shocks. This transition
takes place through an intermediate region where the system exhibits rich
multifractal behavior. This is mainly the region of interest to us. We only
mention in passing the hydrodynamic limit of forcing confined to large scales,
where much work has taken place since that of Polyakov.
In order to make the general framework clear, we give an introduction to
aspects of isotropic, homogeneous turbulence, a description of Kolmogorov
scaling, and, with the help of a simple model, an introduction to the language
of multifractality which is used to discuss intermittency corrections to
scaling.
We continue with a general discussion of the Burgers equation and forcing,
and some aspects of three dimensional turbulence where - because of the
mathematical analogy between equations derived from the Navier-Stokes and
Burgers equations - one can gain insight from the study of the simpler
stochastic Burgers equation. These aspects concern the connection of
dissipation rate intermittency exponents with those characterizing the
structure functions of the velocity field...

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## ‣ Enstrophy growth in the viscous Burgers equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/02/2012
Português

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We study bounds on the enstrophy growth for solutions of the viscous Burgers
equation on the unit circle. Using the variational formulation of Lu and
Doering, we prove that the maximizer of the enstrophy's rate of change is sharp
in the limit of large enstrophy up to a numerical constant but does not
saturate the Poincar\'e inequality for mean-zero 1-periodic functions. Using
the dynamical system methods, we give an asymptotic representation of the
maximizer in the limit of large enstrophy as a viscous shock on the background
of a linear rarefactive wave. This asymptotic construction is used to prove
that a larger growth of enstrophy can be achieved when the initial data to the
viscous Burgers equation saturates the Poincar\'e inequality up to a numerical
constant.
An exact self-similar solution of the Burgers equation is constructed to
describe formation of a metastable viscous shock on the background of a linear
rarefactive wave. When we consider the Burgers equation on an infinite line
subject to the nonzero (shock-type) boundary conditions, we prove that the
maximum enstrophy achieved in the time evolution is scaled as
$\mathcal{E}^{3/2}$, where $\mathcal{E}$ is the large initial enstrophy,
whereas the time needed for reaching the maximal enstrophy is scaled as
$\mathcal{E}^{-1/2} \log(\mathcal{E})$. Similar but slower rates are proved on
the unit circle.; Comment: 40 pages...

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## ‣ On the Convergence of the Convectively Filtered Burgers Equation to the Entropy Solution of the Inviscid Burgers Equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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This document provides a proof that the solutions to the convectively
filtered Burgers equation, will converge to the entropy solution of the
inviscid Burgers equation when certain restrictions are put on the initial
conditions. It does so by first establishing convergence to a weak solution of
the inviscid Burgers equation and then showing that the weak solution is the
entropy solution. Then the results are extended to encompass more general
initial conditions.; Comment: Minor changes and typo corrections

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## ‣ Reduction operators of Burgers equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

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#Mathematical Physics#Mathematics - Analysis of PDEs#Nonlinear Sciences - Exactly Solvable and Integrable Systems#35A30 (Primary) 35C05, 35K59, 35K05 (Secondary)

The solution of the problem on reduction operators and nonclassical
reductions of the Burgers equation is systematically treated and completed. A
new proof of the theorem on the special "no-go" case of regular reduction
operators is presented, and the representation of the coefficients of operators
in terms of solutions of the initial equation is constructed for this case. All
possible nonclassical reductions of the Burgers equation to single ordinary
differential equations are exhaustively described. Any Lie reduction of the
Burgers equation proves to be equivalent via the Hopf-Cole transformation to a
parameterized family of Lie reductions of the linear heat equation.; Comment: 11 pages, minor corrections

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## ‣ Noncommutative Burgers Equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

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#High Energy Physics - Theory#Mathematical Physics#Nonlinear Sciences - Exactly Solvable and Integrable Systems

We present a noncommutative version of the Burgers equation which possesses
the Lax representation and discuss the integrability in detail. We find a
noncommutative version of the Cole-Hopf transformation and succeed in the
linearization of it. The linearized equation is the (noncommutative) diffusion
equation and exactly solved. We also discuss the properties of some exact
solutions. The result shows that the noncommutative Burgers equation is
completely integrable even though it contains infinite number of time
derivatives. Furthermore, we derive the noncommutative Burgers equation from
the noncommutative (anti-)self-dual Yang-Mills equation by reduction, which is
an evidence for the noncommutative Ward conjecture. Finally, we present a
noncommutative version of the Burgers hierarchy by both the Lax-pair generating
technique and the Sato's approach.; Comment: 24 pages, LaTeX, 1 figure; v2: discussions on Ward conjecture, Sato
theory and the integrability added, references added, version to appear in J.
Phys. A

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## ‣ The regularizing effects of resetting in a particle system for the Burgers equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We study the dissipation mechanism of a stochastic particle system for the
Burgers equation. The velocity field of the viscous Burgers and Navier-Stokes
equations can be expressed as an expected value of a stochastic process based
on noisy particle trajectories [Constantin and Iyer Comm. Pure Appl. Math. 3
(2008) 330-345]. In this paper we study a particle system for the viscous
Burgers equations using a Monte-Carlo version of the above; we consider N
copies of the above stochastic flow, each driven by independent Wiener
processes, and replace the expected value with $\frac{1}{N}$ times the sum over
these copies. A similar construction for the Navier-Stokes equations was
studied by Mattingly and the first author of this paper [Iyer and Mattingly
Nonlinearity 21 (2008) 2537-2553]. Surprisingly, for any finite N, the particle
system for the Burgers equations shocks almost surely in finite time. In
contrast to the full expected value, the empirical mean $\frac{1}{N}\sum_1^N$
does not regularize the system enough to ensure a time global solution. To
avoid these shocks, we consider a resetting procedure, which at first sight
should have no regularizing effect at all. However, we prove that this
procedure prevents the formation of shocks for any $N\geq2$...

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## ‣ Local and global well-posedness of the stochastic KdV-Burgers equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/09/2011
Português

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The stochastic PDE known as the Kardar-Parisi-Zhang equation (KPZ) has been
proposed as a model for a randomly growing interface. This equation can be
reformulated as a stochastic Burgers equation. We study a stochastic
KdV-Burgers equation as a toy model for this stochastic Burgers equation. Both
of these equations formally preserve spatial white noise. We are interested in
rigorously proving the invariance of white noise for the stochastic KdV-Burgers
equation. This paper establishes a result in this direction. After smoothing
the additive noise (by less than one spatial derivative), we establish (almost
sure) local well-posedness of the stochastic KdV-Burgers equation with white
noise as initial data. We also prove a global well-posedness result under an
additional smoothing of the noise.; Comment: 39 pages

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