Similarity Solutions of a Generalized Burgers Equation

The similarity method is applied to a generalized Burgers equationwhich has been applied to shock waves and to sound waves. Threedifferent cases of the equation, each allowing a three-parametersymmetry group, are found. The corresponding reduction to anordinary differential equation is given. The Lie algebras ofthe groups are identified with standard types given by Bianchi.     

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.     

Solutions of a Nonhomogeneous Burgers Equation

In this article, we construct solutions of a nonhomogeneous Burgers equation subject to certain unbounded initial profiles. In an interesting study, Kloosterziel [ 1 ] represented the solution of an initial value problem (IVP) for the heat equation, with initial data in , as a series of the self‐similar solutions of the heat equation. This approach quickly revealed the large time behavior for the solution of the IVP. Inspired by Kloosterziel [ 1 ]'s approach, we express the solution of the nonhomogeneous Burgers equation in terms of the self‐similar solutions of a linear partial differential equation with variable coefficients. Finally, we also obtain the large time behavior of the solution of the nonhomogeneous Burgers equation.     

Generalized Burgers Equations Transformable to the Burgers Equation

Using the mappings which involve first‐order derivatives, the Burgers equation with linear damping and variable viscosity is linearized to several parabolic equations including the heat equation, by applying a method which is a combination of Lie’s classical method and Kawamota’s method. The independent variables of the linearized equations are not t, x but z(x, t), τ(t) , where z is the similarity variable. The linearization is possible only when the viscosity Δ(t) depends on the damping parameter α and decays exponentially for large t . And the linearization makes it possible to pose initial and/or boundary value problems for the Burgers equation with linear damping and exponentially decaying viscosity. Bäcklund transformations for the nonplanar Burgers equation with algebraically decaying viscosity are also reported.     

Burgers方程的一类自相似解

利用李群理论中的伸缩变换群,将二阶非线性偏微分方程-Burgers方程化为一类Riccati方程和三类二阶非线性常微分方程,从而Riccati方程和这三类二阶非线性常微分方程给出了Burgers方程的自相似解的表现形式.     

On Generalized Nonlinear Burgers＇ Equation

In this work, we study the similarity analysis of the generalized nonlinear Burgers＇ equation having diffusivity and viscosity as a general functions of the particle velocity. We perform the symmetry analysis and exhibit exact solutions for various forms of the diffusivity and viscosity.     

Exact N-Wave Solutions of Generalized Burgers Equations

Exact N-Wave solutions for the generalized Burgers equation where j, α, β, and γ are nonnegative constants and n is a positive integer, are obtained. These solutions are asymptotic to the (linear) old-age solution for large time and extend the validity of the latter so as to cover the entire time regime starting where the originally sharp shock has become sufficiently thick and the viscous effects are felt in the entire N wave.     

非齐次Burgers方程解的渐近性行为

构造了非齐次Burgers方程的解,方程服从有界和紧致的初始曲线[Kloosterziel RC.J Engrg Math,1990,24(3):213-236],作了一个有趣的探索.将热方程初值问题(L2(R,ex2/2)中有初值)的解,表示为该热方程自相似解的一个级数,Kloosterziel方法立即显示出该初值问题解的渐近性行为.受Kloosterziel方法的启发,根据热方程的自相似解,来表示非齐次Burgers方程的解.最后得到该非齐次Burgers方程解的渐近性特征.     

广义Burgers方程的Legendre-Galerkin Chebyshev-配置方法

本文对广义Burgers方程的Neumann和Robin型边值问题构造了LegendreGalerkinChebyshev-配置方法.Legendre-GalerkinChebyshev-配置方法整体上按LegendreGalerkin方法形成,但对非线性项采用在Chebyshev-Gauss-Lobatto点上的配置法处理.文中给出了方法的稳定性和收敛性分析,获得了按H1-模的最佳误差估计.数值实验证实了方法的有效性.     

Asymptotic N-wave Solutions of the Nonplanar Burgers Equation

In this paper, we construct asymptotic N-wave solutions for the nonplanar Burgers equation as   t →∞  via a balancing argument. These constructed asymptotics are compared with the approximate solutions of the nonplanar Burgers equation obtained by an approach due to Parker ( Acoust. Lett. 4 (1981)). We also present a computationally convenient form for the N-wave solution of the nonplanar Burgers equation modifying Sachdev et al.'s ( Stud. Appl. Math . 103 (1999)) approach. The asymptotic N-wave solutions obtained by balancing argument and modification to Sachdev et al.'s approach are validated by a careful numerical study.     

Exact Linearization and Invariant Solutions of the Generalized Burgers Equation with Linear Damping and Variable Viscosity

The generalized Burgers equation with linear damping and variable viscosity is subjected to Lie's classical method. Five distinct expressions for the variable viscosity are identified. Both the reduced ordinary differential equations and their corresponding Euler-Painlevé transcendents admit first integrals in the form of Bernoulli's equation and are linearized to obtain solutions in closed form.     

一类Fisher方程的行波解

本文讨论了一类广义Fisher方程,得到了它的多个显式行波解.     

一类广义Boussinesq方程的Wronskian行列式解

为了构造非线性孤子方程的Wronskian行列式新解,进一步研究了Wronskian技巧.本文首先给出非线性广义Boussinesq方程的双线性形式,利用Wronskian技巧构造出该非线性方程所满足的一个线性偏微分条件方程组,然后求解该微分条件方程组,得到了广义Boussinesq方程的各种Wronskian行列式解.     

Sensitivity of Solutions to a Parametric Generalized Equation

In the present paper we prove that, under some suitable conditions on multifunctions $C\!: {I\!\!R}^l\rightrightarrows {I\!\!R}^n$ , $F\!:{I\!\!R}^d\times {I\!\!R}^n\rightrightarrows {I\!\!R}^m$ , and $K\!:{I\!\!R}^l\times {I\!\!R}^n\rightrightarrows {I\!\!R}^m$ , the generalized perturbation multifunction $G\!: {I\!\!R}^d\times{I\!\!R}^l\times {I\!\!R}^m\rightrightarrows {I\!\!R}^n$ , of the form $$G(\mu,\lambda,\nu)=\{x\in C(\lambda)\ |\ \nu\in F(\mu,x)+K(\lambda,x)\},$$ is proto-differentiable at (μ, λ, ν) relative to x?∈?G(μ, λ, ν). Moreover, in a special case, where K(λ, x) is a normal cone to C(λ) at x, we also provide sufficient conditions for G(·) to be single-valued on a neighborhood of (μ, λ, ν) and semi-differentiable at (μ, λ, ν).     

非线性发展方程的变系数均衡作用法与变系数Burgers方程的解

利用非线性发展方程的变系数均衡作用法 ,借助计算机符号计算 ,求出变系数 Burgers方程的一种形式的解析解 .     

Large-Time Asymptotics for Periodic Solutions of Some Generalized Burgers Equations

In this paper, we construct large-time asymptotic solutions of some generalized Burgers equations with periodic initial conditions by using a balancing argument. These asymptotics are validated by a careful numerical study. We also show that our asymptotic results agree with the approximate solutions obtained by Parker [1] in certain limits.     

Large-Time Asymptotics for Periodic Solutions of the Modified Burgers Equation

In this paper, we construct large-time asymptotic solution of the modified Burgers equation with sinusoidal initial conditions by using a balancing argument. These asymptotics are validated by a careful numerical study.     

Dhombres型函数方程的连续解

本文用构造的方法给出了Dhombres型函数方程g(x/kg(x))=9(x)~r的连续解,其中k为满足k0(k≠1)的实参数,推广了已有文献关于r=2的结论.     

Analytic Solutions of Klein-gordon Equation and Generalized Schroedinger Equation

Analytic solutions of Klein-gordon equation utt- (uxx uyy) α^2u g(uu^* )u = 0 and generalized Schroedinger equation iut uxx - uyy g( uu^* ) u = 0 are given when g(z) = Aln^2z Blnz C.     

A Support Theorem for a Generalized Burgers SPDE

When the initial condition u ₀ to a parabolic Burgers SPDE (containing a quadratic term) belongs to L ^q [0,1],2q, the trajectories of the solution u(t,x) a.s. belong to the space C([0,T],L ^q [0,1]). We characterize the support of the law of u in this space; the proof is based on an approximation of u by a sequence of stochastic processes obtained by replacing the Brownian sheet by linear adapted interpolations.