The Burgers Equation: Small Perturbations; the Heat Equation

2008 ◽  
pp. 63-71
Keyword(s):  
Heat Equation ◽  
Burgers Equation ◽  
Small Perturbations

A FDM for Burgers’ Equation on Unbounded Domains Using High-Order Artificial Boundary Conditions

2017 ◽  
Vol 865 ◽  
pp. 233-238
Author(s):  
Quan Zheng ◽  
Yu Feng Liu
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Unbounded Domain ◽  
Burgers Equation ◽  
High Order ◽  
Computational Domain ◽  
Artificial Boundary ◽  
Artificial Boundary Conditions ◽  
Second Order Convergence ◽  
The Stability

Burgers’ equation on an unbounded domain is an important mathematical model to treat with some external problems of fluid materials. In this paper, we study a FDM of Burgers’ equation using high-order artificial boundary conditions on the unbounded domain. First, the original problem is converted into the heat equation on an unbounded domain by Hopf-Cole transformation. Thus the difficulty of nonlinearity of Burgers’ equation is overcome. Second, high-order artificial boundary conditions are given by using Padé approximation and Laplace transformation. And the conditions confine the heat equation onto a bounded computational domain. Third, we prove the solutions of the resulting heat equation and Burgers’ equation are both stable. Fourth, we establish the FDM for Burgers’ equation on the bounded computational domain. Finally, a numerical example demonstrates the stability, the effectiveness and the second-order convergence of the proposed method.

scholarly journals The burgers equation with a noisy force and the stochastic heat equation

1994 ◽  
Vol 19 (1-2) ◽  
pp. 119-141 ◽  
Author(s):  
H. Holden ◽  
T. Lindstrøm ◽  
B. øksendal ◽  
J. Ubøe ◽  
T.S. Zhang
Keyword(s):  
Heat Equation ◽  
Burgers Equation ◽  
Stochastic Heat Equation ◽  
The Stochastic Heat Equation

scholarly journals Two Expanding Integrable Models of the Geng-Cao Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Xiurong Guo ◽  
Yufeng Zhang ◽  
Xuping Zhang
Keyword(s):  
Heat Equation ◽  
Lie Algebras ◽  
Burgers Equation ◽  
Integrable Model ◽  
Integrable Models ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Generalized Burgers Equation ◽  
Integrable Couplings ◽  
Integrable Coupling

As far as linear integrable couplings are concerned, one has obtained some rich and interesting results. In the paper, we will deduce two kinds of expanding integrable models of the Geng-Cao (GC) hierarchy by constructing different 6-dimensional Lie algebras. One expanding integrable model (actually, it is a nonlinear integrable coupling) reduces to a generalized Burgers equation and further reduces to the heat equation whose expanding nonlinear integrable model is generated. Another one is an expanding integrable model which is different from the first one. Finally, the Hamiltonian structures of the two expanding integrable models are obtained by employing the variational identity and the trace identity, respectively.

Second-Order Convergence and Unconditional Stability on Crank-Nicolson Scheme for Burgers’ Equation

2013 ◽  
Vol 871 ◽  
pp. 15-20
Author(s):  
Quan Zheng ◽  
Lei Fan ◽  
Guan Ying Sun
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Burgers Equation ◽  
Second Order ◽  
Dirichlet Boundary Conditions ◽  
Second Order Convergence ◽  
Nicolson Scheme ◽  
Homogeneous Dirichlet Boundary Conditions ◽  
Robin Boundary ◽  
Crank Nicolson

In this paper, we study the numerical solution of one-dimensional Burgers equation with non-homogeneous Dirichlet boundary conditions. This nonlinear problem is converted into the linear heat equation with non-homogeneous Robin boundary conditions by Hopf-Cole transformation. The heat equation is discretized by Crank-Nicolson finite difference scheme, and the fourth-order difference schemes for the Robin conditions are combined with the Crank-Nicolson scheme at two endpoints. The proposed method is proved to be second-order convergent and unconditionally stable. The numerical example supports the theoretical results.

On the Cole–Hopf transformation and integration by parts formulae in computational methods within fractional differential equations and fractional optimal control theory

2021 ◽  
pp. 107754632110310
Author(s):  
Ahmed S Hendy ◽  
Mahmoud A Zaky ◽  
José A Tenreiro Machado
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Heat Equation ◽  
Fractional Differential Equations ◽  
Burgers Equation ◽  
Variable Order ◽  
Integer Order ◽  
Order Variable ◽  
Fractional Differential ◽  
Distributed Order

The treatment of fractional differential equations and fractional optimal control problems is more difficult to tackle than the standard integer-order counterpart and may pose problems to non-specialists. Due to this reason, the analytical and numerical methods proposed in the literature may be applied incorrectly. Often, such methods were established for the classical integer-order operators and are then applied directly without having in mind the restrictions posed by their fractional-order versions. It was recently reported that the Cole–Hopf transformation can be used to convert the time-fractional nonlinear Burgers’ equation into the time-fractional linear heat equation. In this article, we show that, unlike integer-order differential equations, employing the Cole–Hopf transformation for reducing the nonlinear time-fractional Burgers’ equation into the time-fractional heat equation is wrong from two different perspectives. Indeed, such a reduction is accomplished using incorrect transcripts of the Leibniz or chain rules. Hence, providing numerical or analytical schemes based on the Cole–Hopf transformation leads to erroneous results for the nonlinear time-fractional Burgers’ equation. Regarding constant-order, variable-order, and distributed-order Caputo fractional optimal problems, we note an inconsistency in the necessary optimality conditions derived in the literature. The transversality conditions were introduced as identical to those for the integer-order case, with a vanishing multiplier at the terminal of the interval. The correct condition should involve a constant-order, variable-order, or distributed-order fractional integral operator. We also deduce that if the control system is defined with a Caputo derivative, then the adjoint equations should be expressed in the Riemann–Liouville sense and vice versa. In fact, neglecting some terms in the integration by parts formulae, during the derivation of the optimality conditions, causes some confusion in the literature.

Resolving controls for exact and approximate controllability of viscous Burgers’ equation: The Green’s function approach

2018 ◽  
Vol 29 (06) ◽  
pp. 1850045 ◽  
Author(s):  
Asatur Zh. Khurshudyan
Keyword(s):  
Heat Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Approximate Controllability ◽  
Burgers Equation ◽  
Sufficient Conditions ◽  
Exact Controllability ◽  
Function Approach ◽  
Dimensional System ◽  
Linear Algebraic Equations

In this paper, we consider a nonlinear control problem for one-dimensional viscous Burgers’ equation associated with a controlled linear heat equation by means of the Hopf–Cole transformation. The control is carried out by the time-dependent intensity of a distributed heat source influencing the heat equation. The set of admissible controls consists of compactly supported [Formula: see text] functions. Using the Green’s function approach, we analyze the possibilities of exact and approximate establishment of a given terminal state for the associated nonlinear Burgers’ equation within a desired amount of time. It is shown that the exact controllability of the associated Burgers’ equation and the heat equation are equivalent. Furthermore, sufficient conditions for the approximate controllability are derived. The set of resolving controls is constructed in both cases. The determination of the resolving controls providing exact controllability is reduced to an infinite-dimensional system of linear algebraic equations. By means of the heuristic method of resolving control determination, parametric hierarchies of solutions providing approximate controllability are constructed. The results of a numerical simulation supporting the theoretical derivations are discussed.

DYNAMICS OF INTERACTING ALGEBRAIC SOLITONS

1995 ◽  
Vol 09 (17) ◽  
pp. 1985-2081 ◽  
Author(s):  
YOSHIMASA MATSUNO
Keyword(s):  
Water Waves ◽  
Evolution Equations ◽  
Burgers Equation ◽  
Soliton Solutions ◽  
Interaction Process ◽  
Exponential Functions ◽  
Small Perturbations ◽  
Mathematical Properties ◽  
Model Equations ◽  
Overtaking Collision

A survey is made which highlights recent topics on the dynamics of algebraic solitons, which are exact solutions to a certain class of nonlinear integrodifferential evolution equations. The model equations that we consider here are the Benjamin-Ono (BO) and its higher-order equations together with the BO-Burgers equation, a model equation for deep-water waves, the sine-Hilbert (sH) equation and a damped sH equation. While these equations have their origin either in physics or in mathematics, each equation exhibits a novel type of algebraic soliton solution and hence its characteristic is worth studying in its own right. After deriving these equations, we are concerned with each equation separately. We first present explicit N-soliton solutions and then summarize related mathematical properties of the equation. Subsequently, a detailed description is given to the interaction process of two algebraic solitons using the pole expansion of the solution. Particular attention is paid to investigating the effects of small perturbations on the overtaking collision of two BO solitons by employing a direct multisoliton perturbation theory. It is shown that the dynamics of interacting algebraic solitons reveal new aspects which have never been observed in the interaction process of usual solitons expressed in terms of exponential functions.

scholarly journals The nonlinear heat equation with state-dependent parameters and its connection to the Burgers' and the potential Burgers' equation

2014 ◽  
Vol 47 (3) ◽  
pp. 7019-7024 ◽  
Author(s):  
Christoph Josef Backi ◽  
Jan Dimon Bendtsen ◽  
John Leth ◽  
Jan Tommy Gravdahl
Keyword(s):  
Heat Equation ◽  
Burgers Equation ◽  
Nonlinear Heat Equation ◽  
State Dependent

scholarly journals Sharp bounds on enstrophy growth in the viscous Burgers equation

2012 ◽  
Vol 468 (2147) ◽  
pp. 3636-3648 ◽  
Author(s):  
Dmitry Pelinovsky
Keyword(s):  
Heat Equation ◽  
Unit Circle ◽  
Time Evolution ◽  
Burgers Equation ◽  
Initial Conditions ◽  
Numerical Results ◽  
Half Period ◽  
Laplace Method ◽  
Sharp Bounds

We use the Cole–Hopf transformation and the Laplace method for the heat equation to justify the numerical results on enstrophy growth in the viscous Burgers equation on the unit circle. We show that the maximum enstrophy achieved in the time evolution is scaled as , where is the large initial enstrophy, whereas the time needed for reaching the maximal enstrophy is scaled as . These bounds are sharp for initial conditions given by odd C 3 functions that are convex on half-period.

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