Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation

In this paper, we implemented the exp-function method for the exact solutions of the fifth order KdV equation and modified Burgers equation. By using this scheme, we found some exact solutions of the above-mentioned equations.     

Application of Exp‐function method to EW‐Burgers equation

In this article, Exp‐function method is used to obtain an exact solution of the equal‐width wave‐Burgers equation (EW‐Burgers). The method is straightforward and concise, and its applications are promising. It is shown that Exp‐function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving EW‐Burgers equation. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

New Bounds for the Inhomogenous Burgers and the Kuramoto-Sivashinsky Equations

We give a substantially simplified proof of the near-optimal estimate on the Kuramoto-Sivashinsky equation from a previous paper of the third author, at the same time slightly improving the result. That result relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the conservative transport nonlinearity acts as a coercive term. It is the proof of the second ingredient that we substantially simplify by proving a modified Kármán-Howarth-Monin identity for solutions of the inhomogeneous inviscid Burgers equation. We show that this provides a new interpretation of recent results obtained by Golse and Perthame.     

Multiple-front solutions for the Burgers equation and the coupled Burgers equations

In this work, multiple-front solutions for the Burgers equation and the coupled Burgers equations are examined. The tanh–coth method and the Cole–Hopf transformation are used. The work highlights the power of the proposed schemes and the structures of the obtained multiple-front solutions.     

分数阶耦合Burgers方程组的同伦摄动解

本文利用同伦摄动法求关于时间Burgers方程组的二阶近似解,为了说明此方法的有效性我们利用Maple 14软件作出了整数阶耦合Burgers方程组的近似解和精确解的图像.结果表明此方法计算量小,避免了对系数的复杂讨论过程并且得出的近似解精确度较高.     

On the solutions of the time-delayed Burgers equation

The time-delayed Burgers equation is introduced and the improved tanh-function method is used to construct exact multiple-soliton and triangular periodic solutions. For an understanding of the nature of the exact solutions that contained the time-delay parameter, we calculated the numerical solutions of this equation by using the Adomian decomposition method and the variational iteration method (IVM) to the boundary value problem.     

A Space-Time Polynomial Collocation Method for Solving 3D Burgers Equations北大核心CSCD

Burgers方程是一类应用广泛的非线性偏微分方程,方程中的非线性项难以处理。该文提出一种新的时空多项式配点法——多项式特解法求解三维Burgers方程。求解过程分为两步:第一步,对三维Burgers方程中的线性导数项(包括时间导数项),求出相应的多项式特解。第二步,将求出的多项式特解作为基函数,对三维Burgers方程中剩余的非线性项进行迭代求解。与时空多项式函数作为基函数对三维Burgers方程进行直接求解相比,该算法简单易行,得到的近似解精度非常高,算法极其稳定,对于教学过程中提高学生的编程能力,加深对高维Burgers方程的理解能力以及Burgers方程的实际应用具有重要意义。     

Wavelet-Taylor Galerkin Method for the Burgers Equation

In this paper, we propose a wavelet-Taylor Galerkin method for the numerical solution of the Burgers equation. In deriving the computational scheme, Taylor-generalized Euler time discretization is performed prior to wavelet-based Galerkin spatial approximation. The linear system of equations obtained in the process are solved by approximate-factorization-based simple explicit schemes, and the resulting solution is compared with that from regular methods. To deal with transient advection-diffusion situations that evolve toward a convective steady state, a splitting-up strategy is known to be very effective. So the Burgers equation is also solved by a splitting-up method using a wavelet-Taylor Galerkin approach. Here, the advection and diffusion terms in the Burgers equation are separated, and the solution is computed in two phases by appropriate wavelet-Taylor Galerkin schemes. Asymptotic stability of all the proposed schemes is verified, and the L_∞ errors relative to the analytical solution together with the numerical solution are reported. AMS subject classification (2000) 65M70     

同伦分析法求解Burgers方程的初边值问题

采用同伦分析法求解了Burgers方程的一初边值问题,得到了它的近似解析解.在不同粘性系数情形下,对近似解与精确解进行了比较,发现在粘性系数不是非常小的情况下,用此方法得到的解析解与精确解符合地很好.     

广义KdV方程与广义Burgers方程的精确解

利用试探函数法和直接积分法构造广义KdV方程与广义Burgers方程的新的精确解.     

Stochastic stability of Burgers equation

The current paper is devoted to stochastic Burgers equation with driving forcing given by white noise type in time and periodic in space. Motivated by the numerical results of Hairer and Voss, we prove that the Burgers equation is stochastic stable in the sense that statistically steady regimes of fluid flows of stochastic Burgers equation converge to that of determinstic Burgers equation as noise tends to zero.     

An improvised collocation algorithm with specific end conditions for solving modified Burgers equation

In this work, numerical solution of nonlinear modified Burgers equation is obtained using an improvised collocation technique with cubic B‐spline as basis functions. In this technique, cubic B‐splines are forced to satisfy the interpolatory condition along with some specific end conditions. Crank–Nicolson scheme is used for temporal domain and improvised cubic B‐spline collocation method is used for spatial domain discretization. Quasilinearization process is followed to tackle the nonlinear term in the equation. Convergence of the technique is established to be of order O(h⁴ + Δt²) . Stability of the technique is examined using von‐Neumann analysis. L₂ and L_∞ error norms are calculated and are compared with those available in existing works. Results are found to be better and the technique is computationally efficient, which is shown by calculating CPU time.     

分数阶Burgers方程的有限元计算(英文)

建立了一维和二维分数阶Burgers方程的有限元格式.时间分数阶导数使用L1方法离散,空间方向使用有限元方法离散.通过选择合适的基函数,将离散后的方程转化成一个非线性代数方程组,并应用牛顿迭代方法求解.数值实验显示出了方法的有效性.     

Symbolic computation of solutions for a forced Burgers equation

In this paper we give exact solutions for a forced Burgers equation. We make use of the generalized Cole-Hopf transformation and the traveling wave method.     

广义KdV—Burgers方程的势对称和不变解

用微分形式的吴方法讨论了广义KdV—Burgers方程不同系数情况下的势对称,并且利用这些对称求得了相应的不变解,这些解对进一步研究广义KdV—Burgers方程所描述的物理现象具有重要意义．     

Sliding mode control of the forced generalized Burgers equation

^** Email: smaoui{at}mcs.sci.kuniv.edu.kw This paper deals with the sliding mode control (SMC) of theforced generalized Burgers equation via the Karhunen-Loève(K-L) Galerkin method. The decomposition procedure of the K-Lmethod is presented to illustrate the use of this method inanalysing the numerical simulations data which represent thesolutions of the forced generalized Burgers equation for viscosityranging from 1 to 100. The K-L Galerkin projection is used asa model reduction technique for non-linear systems to derivea system of ordinary differential equations (ODEs) that mimicsthe dynamics of the forced generalized Burgers equation. Thedata coefficients derived from the ODE system are then usedto approximate the solutions of the forced Burgers equation.Finally, static and dynamic SMC schemes with the objective ofenhancing the stability of the forced generalized Burgers equationare proposed. Simulations of the controlled system are givento illustrate the developed theory.     

Distributed Control Problems for the Burgers Equation

In this work a distributed optimal control problem for time-dependent Burgers equation is analyzed. To solve the nonlinear control problems the augmented Lagrangian-SQP technique is used depending upon a second-order sufficient optimality condition. Numerical test examples are presented.     

A variety of (3 + 1)‐dimensional Burgers equations derived by using the Burgers recursion operator

A new variety of (3 + 1)‐dimensional Burgers equations is presented. The recursion operator of the Burgers equation is employed to establish these higher‐dimensional integrable models. A generalized dispersion relation and a generalized form for the one kink solutions is developed. The new equations generate distinct solitons structures and distinct dispersion relations as well. Copyright © 2014 John Wiley & Sons, Ltd.     

Generalized Burgers equation with rough transport noise

We introduce a new technique for studying well posedness and energy estimates for evolution equations with a rough transport term. The technique is based on finding suitable space–time weight functions for the equations at hand. As an example we study the well posedness of the generalized viscous Burgers equation perturbed by a rough path transport noise.     

Remarks on global controllability for the Burgers equation with two control forces

In this paper we deal with the viscous Burgers equation. We study the exact controllability properties of this equation with general initial condition when the boundary control is acting at both endpoints of the interval. In a first result, we prove that the global exact null controllability does not hold for small time. In a second one, we prove that the exact controllability result does not hold even for large time.