变系数广义KdV—Burgers方程的精确类孤子解

利用一种函数变换将变系数广义KdV—Burgers方程约化为非线性常微分方程（NLODE）．并由此NLODE出发获得变系数广义KdV—Burgers方程的若干精确类孤子解。由此可见,用这种方法还可以求解一大类变系数非线性演化方程。     

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.     

利用多尺度分析方法求解KdV方程和KdV—Burgers方程的格子Boltzmann模型

本文建立了用格点法解一般偏微分方程(PDE)的理论框架,构造出求解KdV方程及KdV—Burgers方程的三速格子BGK模型。引进三种时间尺度,利用多尺分析求出Boltzmann演化方程的平衡分布函数。     

求解 Burgers 方程的决定方程的两种方法研究

论文研究了两种求解偏微分方程的决定方程的方法,一种是运用向量场及其延拓方法,另一种是通过符号计算软件 maple 自动求解软件包。论文以 Burgers 方程为例,证明两种方法得到的结果相同。但运用 maple 自动求解软件包能够避免复杂的代数计算,提高计算的速度与准确率,适用于求解复杂的高阶非线性偏微分方程。     

非线性偏微分方程数值求解的自适应方法研究

对小波理论在偏微分方程数值求解中的应用进行深入研究的基础上,提出了一种自适应求解非线性偏微分方程的算法——小波最优有限差分法。并以非线性Burgers方程为例,分别用小波最优有限差分法和直线法对它进行数值求解,显示了小波最优有限差分法在数值求解非线性问题时的自适应性、高效性和可行性。     

用余弦微分求积法数值求解KdV—Burgers方程

采用余弦微分求积法（CDQM）对（1＋1）维非线性KdV—Burgers方程进行了数值求解．结果表明,所得数值解与方程的精确解相比具有明显的高精度且稳定性高,相对于其他常用方法,且公式简单,使用方便;计算量小,时间复杂性好．     

浅析孤子求解的微扰理论

实际问题中孤子方程通常并不以标准形式出现,而是含有阻尼项、增益项、三阶色散项等一些较小的附加项,将这些附加项当作微扰来处理进而求解各种非线性偏微分方程,发展出了多种孤子微扰理论。本文结合KdV方程详细介绍了两种应用范围最广的基于逆散射变换(IST)的微扰理论和基于线性编微分方程理论的直接微扰理论。     

Chebyshev谱-Euler混合方法求解一类非线性Burgers方程

将Chebyshev谱方法与Euler方法相结合,对一类非线性Burgers方程进行数值求解,通过数值模拟将其与有限差分法和粒子无网格线混合格式MPS-MAFL方法进行了比较,结果表明这种方法对于求解非线性Burgers方程具有较好的效果.     

基于有限元方法的极小曲面造型

讨论极小曲面方程的求解。极小曲面方程是一个高度非线性的二阶椭圆偏微分方程，求解十分困难。该文基于有限元方法，使用一个简单而有效的线性化策略，将问题转化为一系列线性问题，从而大大简化了求解过程。数值结果表明该方法简单有效，能产生合理的结果。     

基于快速汉克尔变换数值求解波方程

1．引言波方程是三大类偏微分方程中的一大类方程，即所谓双曲型偏微分方程，在解决实际问题时极为常见，会经常碰到．比如激光和微波在线性或非线性介质中的传播问题、大气中的声学问题等都要用到波方程．虽然对波方程的数值求解已经研究得很多，但在实际应用中有不同的具体情况，因此仍有必要对方程数值求解方法进行探索．差分法和有限元法是数值求解偏微分方程的两种常用方法，对波方程也不例外．在实际应用中，首先考虑到这两种方法，但显式差分格式对步长有较苛刻的要求，而有限元法是比较复杂的，本征模式（拉盖尔一高斯模）展开法对…     

A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear partial differential equation (PDE) to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order.

A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1+1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved.

The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory.     

Numerical solution of nonlinear system of Klein–Gordon equations by cubic B-spline collocation method

A technique to approximate the solutions of nonlinear Klein–Gordon equation and Klein–Gordon-Schrödinger equations is presented separately. The approach is based on collocation of cubic B-spline functions. The above-mentioned equations are decomposed into a system of partial differential equations, which are further converted to an amenable system of ODEs. The obtained system has been solved by SSP-RK54 scheme. Numerical solutions are presented for five examples, to show the accuracy and usefulness of proposed approach. The approximate solutions of both the equations are computed without using any transformation and linearization. The technique can be applied with ease to solve linear and nonlinear PDEs and also reduces the computational work.     

Homotopy perturbation method for fractional Fornberg–Whitham equation

This article presents the approximate analytical solutions to solve the nonlinear Fornberg–Whitham equation with fractional time derivative. By using initial values, the explicit solutions of the equations are solved by using a reliable algorithm like homotopy perturbation method. The fractional derivatives are taken in the Caputo sense. Numerical results show that the HPM is easy to implement and accurate when applied to time-fractional PDEs.     

A Level Set Framework for Capturing Multi-Valued Solutions of Nonlinear First-Order Equations

We introduce a level set method for the computation of multi-valued solutions of a general class of nonlinear first-order equations in arbitrary space dimensions. The idea is to realize the solution as well as its gradient as the common zero level set of several level set functions in the jet space. A very generic level set equation for the underlying PDEs is thus derived. Specific forms of the level set equation for both first-order transport equations and first-order Hamilton-Jacobi equations are presented. Using a local level set approach, the multi-valued solutions can be realized numerically as the projection of single-valued solutions of a linear equation in the augmented phase space. The level set approach we use automatically handles these solutions as they appear     

On the application of the Cole-Hopf transformation to hyperbolic equations based on second-sound models

We point out and examine two nonlinear, hyperbolic equations, both of which arise in kinematic-wave theory, that can be solved exactly using a conditional application of the Cole-Hopf transformation. Both of these equations are based on flux relations that were originally proposed as models of thermal wave phenomena, also known as second-sound. We then show how this method can be extended and used to obtain a particular type of exact solution to a class of nonlinear, hyperbolic PDEs.     

非对称强非线性振动特征分析

提出了构造一类非线性振子解析逼近周期解的的初值变换法.用Ritz-Galerkin法,将描述动力系统的二阶常微分方程,化为以振幅、角频率和偏心距为独立变量的不完备非线性代数方程组;关键是考虑初值变换,增加补充方程,构成了以角频率、振幅和偏心距为变量的完备非线性代数方程组.作为例子利用初值变换法求解了相对论修正轨道方程的六种分岔周期解.给出了非对称振动的幅频曲线和偏频(偏心距与角频率的关系)曲线.发现了固有角频率漂移现象.     

The Riccati equation with variable coefficients expansion algorithm to find more exact solutions of nonlinear differential equations

In this paper based on a system of Riccati equations with variable coefficients, we present a new Riccati equation with variable coefficients expansion method and its algorithm, which are direct and more powerful than the tanh-function method, sine-cosine method, the generalized hyperbolic-function method and the generalized Riccati equation with constant coefficient expansion method to construct more new exact solutions of nonlinear differential equations in mathematical physics. A pair of generalized Hamiltonian equations is chosen to illustrate our algorithm such that more families of new exact solutions are obtained which contain soliton-like solution and periodic solutions. This algorithm can also be applied to other nonlinear differential equations.     

Lie symmetry analysis and conservation law of variable-coefficient Davey–Stewartson equation

A variable-coefficient Davey–Stewartson (vcDS) equation is investigated in this paper. Infinitesimal generators and symmetry groups are presented by the Lie group method, and the optimal system is presented with adjoint representation. Based on the optimal system, similarity reductions to partial differential equations (PDEs) are obtained, then some PDEs are reduced to ordinary differential equations (ODEs) by two-dimensional subalgebras, and the similarity solutions are provided, including periodic solutions and elliptic function solutions. With Lagrangian, it is shown that vcDS is nonlinearly self-adjoint. Furthermore, based on nonlinear self-adjointness, conservation laws for vcDS equation are derived.     

Searching for Painlevé integrable conditions of nonlinear PDEs with constant parameters using symbolic computation

Based on the Kruskal simplification for the WTC method, a generalized algorithm is devised to establish P-integrable (Painlevé integrable) conditions for nonlinear PDEs with multiple constant parameters. The generalized algorithm fully considers the impact of parameter coefficients upon every step of the Painlevé test. For parametric constraints obtained in the resonance analysis and verification of resonant conditions, the original equations should be regarded as a new system and repeating the test again. For illustration, we apply the generalized algorithm to coupled Schrödinger-Boussinesq equations. Based on the generalized algorithm, a Maple package SPIC is presented, which attributes to derive P-integrable conditions for given nonlinear PDEs with general forms. The higher order water wave equation and coupled KdV equations are selected to illustrate the effectiveness of our package. As a result, some P-integrable conditions are in agreement with the known results, in addition, several new P-integrable models are first given.