Dynamics in spectral solutions of Burgers equation

For low values of the viscosity coefficient, Burgers equation can develop sharp discontinuities, which are difficult to simulate in a computer. Oscillations can occur by discretization through spectral collocation methods, due to Gibbs phenomena. Under a dynamic point of view, these instabilities are related to bifurcations arising to the discretized equation. For different values of discretized points, herein a study is performed of the dynamics and bifurcations occurring in the spectral solutions of Burgers equation with symmetry. Several phenomena are observed, from limit cycles, strange attractors to the presence of bistability with two periodic attractors, with a periodic attractor and a strange attractor and with two strange attractors. Also, other stable equilibrium points can occur, diverse from the ones corresponding to the solution of Burgers equation.     

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.     

Chebyshev Wavelet collocation method for solving generalized Burgers–Huxley equation

In this paper, new and efficient numerical method, called as Chebyshev wavelet collocation method, is proposed for the solutions of generalized Burgers–Huxley equation. This method is based on the approximation by the truncated Chebyshev wavelet series. By using the Chebyshev collocation points, algebraic equation system has been obtained and solved. Approximate solutions of the generalized Burgers–Huxley equation are compared with exact solutions. These calculations demonstrate that the accuracy of the Chebyshev wavelet collocation solutions is quite high even in the case of a small number of grid points. Copyright © 2015 John Wiley & Sons, Ltd.     

一维Burgers方程和KdV方程的广义有限谱方法

给出了高精度的广义有限谱方法．为使方法在时间离散方面保持高精度,采用了Adams-Bashforth 预报格式和Adams-Moulton校正格式,为了避免由Korteweg-de Vries(KdV)方程的弥散项引起的数值振荡, 给出了两种数值稳定器．以Legendre多项式、Chebyshev多项式和Hermite多项式为基函数作为例子,给出的方法与具有分析解的Burgers方程的非线性对流扩散问题和KdV方程的单孤独波和双孤独波传播问题进行了比较,结果非常吻合．     

Numerical solutions for convection–diffusion equation using El-Gendi method

The method of El-Gendi [El-Gendi SE. Chebyshev solution of differential integral and integro-differential equations. J Comput 1969;12:282–7; Mihaila B, Mihaila I. Numerical approximation using Chebyshev polynomial expansions: El-gendi’s method revisited. J Phys A Math Gen 2002;35:731–46] is presented with interface points to deal with linear and non-linear convection–diffusion equations.The linear problem is reduced to two systems of ordinary differential equations. And, then, each system is solved using three-level time scheme.The non-linear problem is reduced to three systems of ordinary differential. Each one of these systems is, then, solved using three-level time scheme. Numerical results for Burgers’ equation and modified Burgers’ equation are shown and compared with other methods. The numerical results are found to be in good agreement with the exact solutions.     

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．     

Least-squares spectral element method for non-linear hyperbolic differential equations

The least-squares spectral element method has been applied to the one-dimensional inviscid Burgers equation which allows for discontinuous solutions. In order to achieve high order accuracy both in space and in time a space–time formulation has been applied. The Burgers equation has been discretized in three different ways: a non-conservative formulation, a conservative system with two variables and two equations: one first order linear PDE and one linearized algebraic equation, and finally a variant on this conservative formulation applied to a direct minimization with a QR-decomposition at elemental level. For all three formulations an h/p-convergence study has been performed and the results are discussed in this paper.     

The Chebyshev Spectral Method for Burgers-Like Equations

The Chebyshev polynomials have good approximation properties which are not affected by boundary values. They have higher resolution near the boundary than in the interior and are suitable for problems in which the solution changes rapidly near the boundary. Also, they can be calculated by FFT. Thus they are used mostly for initial-boundary value problems for P.D.E.'s (see [1, 3-4, 6, 8-11]). Maday and Quarterom discussed the convergence of Legendre and Chebyshev spectral approximations to the steady Burgers equation. In this paper we consider Burgers-like equations.$$\begin{cases}∂_iu+F(u)_x-vu_{zx}=0, & -1≤x≤1, 0＜t≤T \\ u (-1,t) =u (1,t) =0, & 0≤t≤T & (0.1)\\ u (x,0) =u_0(x), & -1≤x≤1\end{cases}$$ where $F\in C(R)$ and there exists a positive function $A\in C(R)$ and a constant $p＞1$ such that $$|F(z+y)-F(z)|\leq A(z)(|y|+|y|^p).$$ We develop a Chebyshev spectral scheme and a pseudospectral scheme for solving (0.1) and establish their generalized stability and convergence.     

Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers‐Hilbert equation as a model equation for such waves and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasi‐linear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two‐dimensional inviscid, incompressible fluid flows. Thus, the Burgers‐Hilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers‐Hilbert and asymptotic equations, and show that the asymptotic equation can also be derived by means of a near‐identity transformation. We derive a semiclassical approximation of the asymptotic equation and show that spatially periodic, harmonic traveling waves are linearly and modulationally stable. Numerical solutions of the Burgers‐Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small‐amplitude smooth solutions of the Burgers‐Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation. © 2009 Wiley Periodicals, Inc.     

Analytical analysis for large-amplitude oscillation of a rotational pendulum system

This paper deals with large amplitude oscillation of a nonlinear pendulum attached to a rotating structure. By coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials, the governing differential equation with sinusoidal nonlinearity can be reduced to form a cubic-quintic Duffing equation. The resulting Duffing type temporal problem is solved by an analytic iteration approach. Two approximate formulas for the frequency (period) and the periodic solution are established for small as well as large amplitudes of motion. Illustrative examples are selected and compared to those analytical and exact solutions to substantiate the accuracy and correctness of the approximate analytical approach.     

Stochastic Burgers PDEs with random coefficients and a generalization of the Cole–Hopf transformation

This paper studies forward and backward versions of the random Burgers equation (RBE) with stochastic coefficients. First, the celebrated Cole–Hopf transformation reduces the forward RBE to a forward random heat equation (RHE) that can be treated pathwise. Next we provide a connection between the backward Burgers equation and a system of FBSDEs. Exploiting this connection, we derive a generalization of the Cole–Hopf transformation which links the backward RBE with the backward RHE and investigate the range of its applicability. Stochastic Feynman–Kac representations for the solutions are provided. Explicit solutions are constructed and applications in stochastic control and mathematical finance are discussed.     

无界域上半线性强阻尼波动方程的全离散有理谱逼近

本文运用Chebyshev有理谱方法来讨论半线性强阻尼波动方程.通过建立时间、空间方向全离散的Chebyshev有理谱格式,证明了由此格式所确定的离散算子半群存在整体吸引子,并从理论上建立了在有限时间上近似解的误差估计.     

An adaptive spectral least-squares scheme for the Burgers equation

A least-squares spectral collocation method for the one-dimensional inviscid Burgers equation is proposed. This model problem shows the stability and high accuracy of these schemes for nonlinear hyperbolic scalar equations. Here we make use of a least-squares spectral approach which was already used in an earlier paper for discontinuous and singular perturbation problems (Heinrichs, J. Comput. Appl. Math. 157:329–345, 2003). The domain is decomposed in subintervals where continuity is enforced at the interfaces. Equal order polynomials are used on all subdomains. For the spectral collocation scheme Chebyshev polynomials are employed which allow the efficient implementation with Fast Fourier Transforms (FFTs). The collocation conditions and the interface conditions lead to an overdetermined system which can be efficiently solved by least-squares. The solution technique will only involve symmetric positive definite linear systems. The scheme exhibits exponential convergence where the exact solution is smooth. In parts of the domain where the solution contains discontinuities (shocks) the spectral solution displays a Gibbs-like behavior. Here this is overcome by some suitable exponential filtering at each time level. Here we observe that by over-collocation the results remain stable also for increasing filter parameters and also without filtering. Furthermore by an adaptive grid refinement we were able to locate the precise position of the discontinuity. Numerical simulations confirm the high accuracy of our spectral least-squares scheme.      

COMPOSITE LEGENDRE-LAGUERRE APPROXIMATION IN UNBOUNDED DOMAINS

1. IntroductionMany problems in science aam engineering are set in unbounded domains. There are severalways for their numerical simulatiolls. We may restrict calculations to some bounded domainswith certain artificial boundary conditions. But they induce errors. In particular, they ductshe wave Propagations in revolutionary problems. In opposite, if we use'spectral methods assorted with orthogonal systems of polynomials in Unbounded domains, then we could avoid tabstrouble, e.g., see Mad'ay,…     

Linear Stability of Viscous Roll Waves

The purpose of this paper is to study the linear stability of “viscous” roll waves. These are periodic continuous traveling waves solutions of viscous perturbations of inhomogeneous hyperbolic systems. We first study the scalar case for the Burgers equation and for an inhomogeneous hyperbolic equation. Then we analyze the stability of roll waves, solutions of the shallow water equations with a real viscosity. In both cases, we first analyze the Evans function and compute an asymptotic expansion in the low frequency regime. Under a strong spectral stability condition, we prove the linear stability of viscous roll waves, solutions of the Saint Venant equations, with pointwise estimates on the Green functions.     

Nonautonomous first integrals and general solutions of the KdV‐Burgers and mKdV‐Burgers equations with the source

The method for constructing first integrals and general solutions of nonlinear ordinary differential equations is presented. The method is based on index accounting of the Fuchs indices, which appeared during the Painlevé test of a nonlinear differential equation. The Fuchs indices indicate us the leading members of the first integrals for the origin differential equation. Taking into account the values of the Fuchs indices, we can construct the auxiliary equation, which allows to look for the first integrals of nonlinear differential equations. The method is used to obtain the first integrals and general solutions of the KdV‐Burgers and the mKdV‐Burgers equations with a source. The nonautonomous first integrals in the polynomials form are found. The general solutions of these nonlinear differential equations under at some additional conditions on the parameters of differential equations are also obtained. Illustrations of some solutions of the KdV‐Burgers and the mKdV‐Burgers are given.     

Exact solitary wave solutions for a generalized KdV–Burgers equation

Using the special truncated expansion method, the solitary wave solutions are constructed for the compound Korteweg–de Vries–Burgers (KdVB) equation. Exact and explicit solitary wave solutions for a generalized KdVB equation are obtained by introducing a suitable ansatz equation. The generalized two-dimensional KdVB equation is discussed. Some particular cases of the generalized KdVB equation are solved by using these methods.     

一种改进的截断展开法求非线性发展方程的精确解

给出了一种改进的截断展开法,利用此方法借助于计算机符号计算求得了Burgers方程和浅水长波近似方程组的精确解,其中包括孤子解,并讨论其具体应用.改进后的方法与以前的方法相比能得到方程的更多形式的精确解.所给出的改进的截断展开法也可以用来研究其它非线性发展方程的孤子解,是求非线性发展方程精确解的一种有效的直接方法.     

Self-similar solutions of a Burgers-type equation with quadratically cubic nonlinearity

Self-similar solutions are found for a quadratically cubic second-order partial differential equation governing the behavior of nonlinear waves in various distributed systems, for example, in some metamaterials. They are compared with self-similar solutions of the Burgers equation. One of them describing a single unipolar pulse is shown to satisfy both equations. The other self-similar solutions of the quadratically cubic equation behave differently from the solutions of the Burgers equation. They are constructed by matching the positive and negative branches of the solution, so that the function itself and its first derivative are continuous. One of these solutions corresponds to an asymmetric solitary N-wave of the sonic shock type. Self-similar solutions of a quadratically cubic equation describing the propagation of cylindrically symmetric waves are also found.     

Explicit solution and Darboux transformation for a new discrete integrable soliton hierarchy with 4×4 Lax pairs

The Darboux transformation method with 4×4 spectral problem has more complexity than 2×2 and 3×3 spectral problems. In this paper, we start from a new discrete spectral problem with a 4×4 Lax pairs and construct a lattice hierarchy by properly choosing an auxiliary spectral problem, which can be reduced to a new discrete soliton hierarchy. For the obtained lattice integrable coupling equation, we establish a Darboux transformation and apply the gauge transformation to a specific equation and then the explicit solutions of the lattice integrable coupling equation are obtained. Copyright © 2017 John Wiley & Sons, Ltd.