Landau-Ginzburg-Higgs方程的多辛Runge-Kutta方法

非线性波动方程作为一类重要的数学物理方程吸引着众多的研究者,基于Hamilton空间体系的多辛理论研究了Landau-Ginzburg-Higgs方程的多辛算法,讨论了利用Runge-Kutta方法构造离散多辛格式的途径,并构造了一种典型的半隐式的多辛格式,该格式满足多辛守恒律、局部能量守恒律和局部动量守恒律.数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性.     

饱和多孔弹性杆流固耦合动力响应的保结构算法

研究了不可压饱和多孔弹性杆的流固耦合动力响应问题.基于多孔介质理论,根据多孔介质流固混合物动量方程、孔隙流体动量方程及体积分数方程,建立流固耦合不可压饱和多孔弹性杆的轴向振动方程;引入正则变量,构造饱和多孔弹性杆轴向振动方程的广义多辛保结构形式、广义多辛守恒律及广义多辛局部动量误差;采用中点Box离散方法得到轴向振动方程的广义多辛离散格式、广义多辛守恒律数值误差及局部动量数值误差;数值模拟不可压饱和多孔弹性杆的轴向振动过程及流相渗流速度分布,考察了流固两相耦合系数对轴向振动过程及广义多辛守恒律误差和局部动量误差的影响.结果表明,已构造的广义多辛保结构算法具有很高的精确性和长时间的数值稳定性.     

一类强色散DGH方程的多辛普雷斯曼格式

DGH方程作为一类重要的非线性水波方程有着许多广泛的应用前景.基于Hamilton系统的多辛理论研究了一类强色散DGH方程的数值解法,利用多辛普雷斯曼方法构造了一种典型的半隐式的多辛格式.分析了该格式的局部能量和动量守恒律误差,并给出了数值算例.数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性.     

Sine-Gordon方程的多辛Leap-frog格式

非线性发展方程由于具有多种形式的解析解而吸引着众多的研究者,借助多辛保结构理论研究了Sine-Gordon方程的多辛算法．利用Hamilton变分原理,构造出了sine-Gordon方程的多辛格式;采用显辛离散方法得到了Leap-frog多辛离散格式,该格式满足多辛守恒律;数值结果表明leap-frog多辛离散格式能够精确地模拟sine-Gordon方程的孤子解和周期解,模拟结果证实了该离散格式具有良好的数值稳定性．     

膜自由振动的多辛方法

基于Hamilton空间体系的多辛理论研究了膜自由振动问题,讨论了构造复合离散多辛格式的方法,并构造了一种典型的9×3点半隐式的多辛复合离散格式,该格式满足多辛守恒律、能量守恒律和动量守恒律．数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性．     

不可压饱和多孔弹性杆动力响应的多辛方法

研究了不可压饱和多孔弹性杆的一维动力响应问题.基于多孔介质理论,在流相和固相微观不可压、固相骨架小变形的假定下,建立了不可压流体饱和多孔弹性杆一维轴向动力响应的数学模型.利用Hamilton空间体系的多辛理论,构造了不可压饱和多孔弹性杆轴向振动方程的多辛形式及其多种局部守恒律.采用中点Box离散方法得到轴向振动方程的多辛离散格式和局部能量守恒律以及局部动量守恒律的离散格式;数值模拟了不可压饱和多孔弹性杆的轴向振动过程,记录了每一时间步的局部能量数值误差和局部动量数值误差.结果表明,已构造的多辛离散格式具有很高的精确性和较长时间的数值稳定性,这为解决饱和多孔介质的动力响应问题提供了新的途径.     

MKdV方程的多辛Fourier拟谱方法

基于谱微分矩阵方法,给出MKdV方程的多辛Fourier拟谱格式及其相应多辛离散守恒律,证明了它等价于通常的Fourier拟谱格式．数值结果表明,格式对于长时间计算具有稳定性与高精度．     

广义非线性Schroedinger方程的一个新的守恒差分格式

对广义非线性Schroedinger方程提出了一种新的差分格式。揭示了该差分格式满足两个守恒律，并证明该格式的收敛性和稳定性．数值实验结果表明，新的差分格式优于Crank-Nicolson格式以及Zhang Fei等人提出的格式。     

三维涡度方程双向周期问题的拟谱─差分解法

本文构造了三维涡度方程双向周期问题的Ｆｏｕｒｉｅｒ拟谱─差分格式，其数值解满足半离散守恒律．文中分析了格式的广义稳定性和收敛性．数值例子表明这类格式的优越性．     

二维阻尼非线性sine-Gordon方程的共形多辛Fourier拟谱格式

为二维阻尼非线性sine-Gordon方程构造了一个新的共形多辛Fourier拟谱格式.基于原系统的共形多辛哈密尔顿形式,首先在时间和空间方向上分别用辛中点和Fourier拟谱方法进行离散,得到一个全离散格式.随后证明了构造的格式保持离散的共形多辛守恒律.最后数值实验验证了格式的有效性.     

A NEW MULTI-SYMPLECTIC SCHEME FOR NONLINEAR “GOOD” BOUSSINESQ EQUATION

The Hamiltonian formulations of the linear “good“ Boussinesq (L.G.B.) equation and the multi-symplectic formulation of the nonlinear “good“ Boussinesq (N.G.B.) equation are considered. For the multi-symplectic formulation, a new fifteen-point difference scheme which is equivalent to the multi-symplectic Preissmann integrator is derived. We also present numerical experiments, which show that the symplectic and multisymplectic schemes have excellent long-time numerical behavior.     

非线性Pochhammer-Chree方程的多辛格式

提出非线性Pochhammer—Chree方程的多辛形式,进而得到一个等价于中心Preissmann积分的15点多辛格式．数值例子表明：多辛格式具有良好的长时间数值行为。     

Multi-soliton solution,rational solution of the Boussinesq–Burgers equations

In this paper we consider the Boussinesq–Burgers equations and establish the transformation which turns the Boussinesq–Burgers equations into the single nonlinear partial differential equation, then we obtain an auto-Bäcklund transformation and abundant new exact solutions, including the multi-solitary wave solution and the rational series solutions. Besides the new trigonometric function periodic solutions are obtained by using the generalized tan h method.     

A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation

In this paper, we employ the boundary-only meshfree method to find out numerical solution of the classical Boussinesq equation in one dimension. The proposed method in the current paper is a combination of boundary knot method and meshless analog equation method. The boundary knot technique is an integration free, boundary-only, meshless method which is used to avoid the known disadvantages of the method of fundamental solution. Also, we use the meshless analog equation method to replace the nonlinear governing equation with an equivalent nonhomogeneous linear equation. A predictor-corrector scheme is proposed to solve the resulted differential equation of the collocation. The numerical results and conclusions are obtained for both the ‘good’ and the ‘bad’ Boussinesq equations.     

Finite difference scheme of a model for nonlinear wave-wave interaction in ionic media

In the paper, the coupled 1D nonlinear Schr?dinger system (CNLS) is considered as the model equation for wave-wave interaction in ionic media. A finite difference scheme is derived for the model equations. A new six-point scheme, which is equivalent to the multi-symplectic integrator, is derived. The numerical simulation is also presented for the model equations.     

A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor's formula

In this article, a novel numerical method is proposed for nonlinear partial differential equations with space- and time-fractional derivatives. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor's formula. The fractional derivatives are considered in the Caputo sense. Several illustrative examples are given to demonstrate the effectiveness of the present method. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. Results also show that the numerical scheme is very effective and convenient for solving nonlinear partial differential equations of fractional order.     

Solving nonlinear fractional partial differential equations using the homotopy analysis method

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2,2), Burgers, BBM‐Burgers, cubic Boussinesq, coupled KdV, and Boussinesq‐like B(m,n) equations with initial conditions, which are introduced by replacing some integer‐order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer‐order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Nonclassical symmetries of a class of nonlinear partial differential equations with arbitrary order and compatibility

In this paper, we show that for a class of nonlinear partial differential equations with arbitrary order the determining equations for the nonclassical reduction can be obtained by requiring the compatibility between the original equation and the invariant surface condition. The nonlinear wave equation and the Boussinesq equation all serve as examples illustrating this fact.