Burgers方程的精确解

引入一个变换,将二阶非线性偏微分方程—Burgers方程降阶为一阶的非线性方程,再直接求解该方程,得出了Burgers方程精确解的新形式,并与已有结果完全吻合.这种方法也适合于求解其他非线性偏微分方程.     

广义Burgers方程的对称分类及其约化

利用李群方法对广义Burgers方程ut+f(x,t)(ux-uxx)=0的对称分类及其约化作具体讨论,其中f是关于自变量x,u的光滑函数,得到了f(x,t)的八种分类对称及相应的约化方程.该结果对于广义Burgers方程精确解的研究有重要意义.     

Burgers方程的精确孤立波解的符号计算

借助于符号计算Maple,给出了一种构造非线性波动方程行波解的直接代数方法,该方法的主要特点是充分利用Riccati方程.使用此方法得到Burgers方程的多组精确行波解,其中包括一些新的孤立波解,这种方法也适用于求解其它的非线性波动方程（组）.     

Traveling-wave solutions for a discrete Burgers equation with nonlinear diffusion

We construct a nonstandard finite difference (NSFD) scheme for a Burgers type partial differential equation (PDE) for which the diffusion coefficient has a linear dependence on the dependent variable. After a study of this PDE's traveling-wave solutions, we examine the corresponding properties of the NSFD construction. Our work demonstrates the dynamic consistency of the discretization.     

Asymptotics of the solutions of the generalized Hutchinson equation

The problem of the behavior of solutions of the Hutchinson equation and its generalizations is considered. Results regarding the estimation of the domain of the global stability of the positive equilibrium state in the parameter space are obtained. The problems of the existence, stability, and asymptotics of a slowly oscillating periodic solution are approached in the basic propositions. The problem of the dynamic properties of the system of ordinary differential equations that describes the well-known Belousov-Zhabotinsky reaction is considered as an application of the newly developed asymptotic methods.     

Numerical solutions of Burgers’ equation with a large Reynolds number

In this article the exact solution of Burgers’ equation represented as an infinite series is transformed into a simpler form involving the elliptic function? ₃(υ, q). To evaluate? ₃(υ, q), we use the Jacobi Imaginary Transformation. It is made clear that the solutions obtained by the proposed approach are numerically stable and precise.     

广义BBM方程的无穷序列新解

利用第二种椭圆方程的已知解与解的非线性叠加公式,构造了广义BBM方程的由Jacobi椭圆函数解、双曲函数和三角函数组成的无穷序列新解.     

广义Fitzhugh Nagumo方程的无穷序列新解

利用辅助方程的几种结论,构造了广义Fitzhugh-Nagumo方程的多种无穷序列新解.步骤一,利用函数变换与首次积分,给出了辅助方程的新解、Bcklund变换和解的非线性叠加公式.步骤二,通过函数变换,将广义Fitzhugh-Nagumo方程的求解问题转化为非线性常微分方程的求解问题.步骤三,利用符号计算系统Mathematica与辅助方程的几种结论,构造了广义Fitzhugh-Nagumo方程的多种无穷序列新解.     

Wick类型的随机广义Kdv方程的精确解

在Kondratiev分布空间(S)-1中通过埃尔米特变换和Painleve′分析导出了Wick-类型的随机广义Kdv方程的Backlund变换,并且把Wick-类型的随机广义Kdv方程变成广义系数Kdv-方程,再利用Backlund变换求出广义系数Kdv方程的精确解,最后通过埃尔米特逆变换求出随机广义Kdv方程在系数取不同白色噪音泛函条件下的精确解.     

New solitary wave solutions for generalized regularized long-wave equation

In this paper, the sine–cosine and the tan h methods have been used to obtain solutions of the generalized regularized long-wave equation. New solitary and periodic solutions are formally derived. The change of parameters, that will drastically change the characteristics of the equation, is examined.     

非对称NNV方程新的广义孤波解和周期解

指数函数法是求解非线性发展方程的一种简单有效的方法,利用该方法并借助Mathematica软件的符号计算功能求解了非对称NNV方程,得到了新的孤波解和周期解。     

Unified parametrization for the solutions to the polynomial diophantine matrix equation and the generalized Sylvester matrix equation

The polynomial Diophantine matrix equation and the generalized Sylvester matrix equation are important for controller design in frequency domain linear system theory and time domain linear system theory, respectively. By using the so-called generalized Sylvester mapping, right coprime factorization and Bezout identity associated with certain polynomial matrices, we present in this note a unified parametrization for the solutions to both of these two classes of matrix equations. Moreover, it is shown that solutions to the generalized Sylvester matrix equation can be obtained if solutions to the Diophantine matrix equation are available. The results disclose a relationship between the polynomial Diophantine matrix equation and generalized Sylvester matrix equation that are respectively studied and used in frequency domain linear system theory and time domain linear system theory.     

Auto-Bäcklund transformation and exact solutions of the generalized variable-coefficient Kadomtsev-Petviashvili equation

Based on the idea of the homogeneous balance (HB) method, an auto-Bäcklund transformation (BT) to the generalized variable-coefficient Kadomtsev-Petviashvili (GvcKP) equation is obtained with symbolic computation. By the use of the auto-BT and the ε-expansion method, we can obtain a soliton-like solution including N-solitary wave of the GvcKP equation. Especially, we get a soliton-like solution including two-solitary wave as an illustrative example in detail. Since the cylindrical KP (cKP) equation, the generalized cKP (GcKP) equation and the spherical KP (SKP) equation are all special cases of the GvcKP equation, we can also obtain the corresponding results of these equations respectively.     

改进同伦分析方法及推广Kuramoto Sivashinsky方程的近似解

首先介绍了带有两个辅助参数的改进同伦分析方法,然后用该方法得到了推广Kuramoto-Sivashin-sky方程的同伦近似解.所得近似解与精确孤立波解进行比较,发现本文得到的近似解更有效地逼近真实解.因为该解包含了两个辅助参数,所以能够更有效地调节和控制其收敛区域和速度.研究表明带有两个辅助参数的改进同伦分析方法对复杂非线性系统的研究更有它的优点.     

A lumped Galerkin method for solving the Burgers equation

A numerical solution of the one-dimensional Burgers equation is obtained using a lumped Galerkin method with quadratic B-spline finite elements. The scheme is implemented to solve a set of test problems with known exact solutions. Results are compared with published numerical and exact solutions. The proposed scheme performs well. A linear stability analysis shows the scheme to be unconditionally stable.     

Notes on “The new exact solitary and multi-soliton solutions for the (2+1)-dimensional Zakharov–Kuznetsov equation”

In a recent paper referred to in the title, the author investigated a (2+1)-dimensional Zakharov–Kuznetsov (ZK) equation and claimed that with the aid of the coupled Burgers’ equations, a new multi-soliton solution were formally generated. We argue that the generated N-soliton solutions in the case of $N\ge 2$ is incorrect and is not admitted by the original (2+1)-dimensional ZK equation.     

On the exact and numerical solution 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 to the boundary value problem.     

Lump-type solutions and lump solutions for the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation

In this paper, a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation is investigated. Lump-type solutions and lump solutions are obtained with aid of symbolic computation via Hirota bilinear method and the ansatz technique. By taking the function $f$ in the Hirota bilinear form of the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation as the general quadratic polynomial function, a kind of lump-type solution which contains eleven parameters with six arbitrary independent parameters and two non-zero conditions is obtained. Lump solutions are found from the lump-type solutions via taking a special set of parameters, and the motion track of the lump is also described both theoretically and graphically.     

Numerical solutions of two dimensional Sobolev and generalized Benjamin–Bona–Mahony–Burgers equations via Haar wavelets

This work addresses, numerical method based on Haar wavelets and finite differences to solve two dimensional linear, nonlinear Sobolev and non-linear generalized Benjamin–Bona–Mahony–Burgers (NGBBMB) equations. The temporal part is discretized using finite differences while spatial part is approximated by two dimensional Haar wavelets. With this strategy, computing solution of two dimensional PDEs reduces to computing solution of linear system of algebraic equations. Collocation approach is then applied to determine the wavelet coefficients from linear system. This paper shows that two dimensional Haar wavelets are suitable and effective for two dimensional linear and non-linear PDEs. For validation of the proposed scheme different problems have been solved and error norms $L_{\infty },L_2$ are computed. Computation verifies that current scheme has good outcome.