Landau-Lifshitz方程的群不变解和守恒律

利用经典李群方法得到了Landau-Lifshitz方程不变群的无穷小生成元,验证其对换位运算构成一个七维的李代数,得到了对应的群不变解,建立了Landau-Lifshit,z新解和旧解之间的关系.同时利用对称和共轭方程组求得了Landau-Lifshitz方程的守恒律.     

可渗透壁面上Falkner-Skan磁流体动力学流动的近似解

研究了可渗透壁面上Falkner-Skan磁流体动力学（MHD）边界层流动问题．利用结合了微分变换法（DTM）和Padé近似的DTM-Padé方法,得到了边界层问题的近似解和壁摩擦因数值．通过建立一个迭代程序,边界层问题的近似解被表示为幂级数的形式,而且以图和表形式对不同参数下的近似解结果与打靶法得到的数值结果进行了对比,近似解和数值解结果高度吻合,从而验证了所得问题近似解和结论的可靠性和有效性．并且,对求得的边界层问题近似解结果进行了讨论,分析了不同物理参数对边界层流动的影响．     

特殊坐标系中特殊非牛顿流边界层方程的相似解

给出了在一个特殊坐标系中三阶流体的二维定常运动方程组．该坐标系中由无粘流体的势流确定，即以环绕任意物体的非粘性流动的流线为Ф-坐标，速度势线为ψ-坐标，构成正交曲线坐标系．结果表明，边界层方程与浸没在流体中的物体的形状无关．第一次近似假定第二梯度项与粘性项和第三梯度项相比，可以忽略不计．第二梯度项的存在，将防碍第三梯度流相似解的比例变换的导出．利用李群方法计算了边界层方程的无穷小生成元．将边界层方程组变换为常微分方程组．利用Runge-Kutta法结合打靶技术求解了该非线性微分方程组的数值解．     

非线性扰动Klein-Gordon方程初值问题的渐近理论

在二维空间中研究一类非线性扰动Klein-Gordon方程初值问题解的渐近理论. 首先利用压缩映象原理,结合一些先验估计式及Bessel函数的收敛性,根据Klein-Gordon方程初值问题的等价积分方程,在二次连续可微空间中得到了初值问题解的适定性;其次,利用扰动方法构造了初值问题的形式近似解,并得到了该形式近似解的渐近合理性;最后给出了所得渐近理论的一个应用,用渐近近似定理分析了一个具体的非线性Klein-Gordon方程初值问题解的渐近近似程度．     

粘弹性薄板动力响应的边界元方法（Ⅰ）

本文中我们给出了粘弹性薄板动力响应的边界元方法．在Laplace变换区域中，给出了基本解的两种近似方法，运用这些近似基本解建立了边界元方法，再利用改进的Bellman反交换技术，求得问题的解，计算表明该方法具有较高精度和较快收敛性．     

广义相对论磁流体动力学方程及解

本文对磁流体(连续介质,具有无限大电导率)的绝热运动进行分析,得到广义相对论动力学方程组,并在匀加速参考系中求得这组方程的一个解.最后将这个解变换到惯性坐标系中。这些结果对于研究宇宙等离子体将是有用的。     

两类破碎群体平衡方程接受的李群及群不变解

针对难找到破碎群体平衡方程的精确解和解析方法缺乏的问题,研究两类积分-偏微分方程(破碎群体平衡方程)接受的李群、群不变解、约化积分-常微分方程及精确解.首先采用伸缩变换李群分析方法探寻积分-偏微分方程接受的李群.其次将积分-偏微分方程转化为纯偏微分方程,运用经典李群分析方法计算纯偏微分方程接受的李群.然后利用改进了的李群分析方法结合伸缩变换群和经典李群分析方法获得的结果确定积分-偏微分方程接受的李群.最后找到了积分-偏微分方程接受的李群,给出了积分-偏微分方程的约化积分-常微分方程、群不变解及显式精确解,分析了部分解的动力学行为性质及特征.     

一类非线性发展方程组的定解问题

将不可压缩的广义neo-Hookean材料组成的超弹性圆柱壳径向对称运动的数学模型归结为一类非线性发展方程组的初边值问题.利用材料的不可压缩条件和边界条件求得了描述圆柱壳内表面径向运动的二阶非线性常微分方程.给出了微分方程的周期解(即圆柱壳的内表面产生非线性周期振动)的存在条件,讨论了材料参数和结构参数对方程的周期解的影响,并给出了相应的数值模拟.     

李群方法在渗流力学的应用

试图用李群方法来分析流体及渗流的运动规律.对于流形上流体、渗流力学方程的研究,物理空间的流动中的拓扑结构只要具有李群的性质,便可以此来进行流动分析.这是将李群理论直接、直地应用于渗流力学的一种方法.李群方法将众多求解特定类型的渗流微分方程方法统一到共同的概念之下.李群无穷小变换方法为寻找微分方程的闭合形式的解提供的广泛的应用,补充了求解渗流力学方程的数学物理技巧.     

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.     

Fundamental solutions, transition densities and the integration of Lie symmetries

In this paper we present some new applications of Lie symmetry analysis to problems in stochastic calculus. The major focus is on using Lie symmetries of parabolic PDEs to obtain fundamental solutions and transition densities. The method we use relies upon the fact that Lie symmetries can be integrated with respect to the group parameter. We obtain new results which show that for PDEs with nontrivial Lie symmetry algebras, the Lie symmetries naturally yield Fourier and Laplace transforms of fundamental solutions, and we derive explicit formulas for such transforms in terms of the coefficients of the PDE.     

Lie group classifications and exact solutions for two variable-coefficient equations

In this paper, the Lie symmetry analysis and group classifications are performed for two variable-coefficient equations, the hanging chain equation and the bond pricing equation. The symmetries for the two equations are obtained, the exact explicit solutions generated from the similarity reductions are presented. Moreover, the exact analytic solutions are considered by the power series method.     

Similarity analysis of modified shallow water equations and evolution of weak waves

In this paper, we obtain exact solutions to the nonlinear system of partial differential equations (PDEs), describing the one dimensional modified shallow water equations, using invariance group properties of the governing system. Lie group of point symmetries with commuting infinitesimal operators, are presented. The symmetry generators are used for constructing similarity variables which lead the governing system of PDEs to system of ordinary differential equations (ODEs); in some cases, it is possible to solve these equations exactly. A particular solution to the governing system, which exhibits space-time dependence, is used to study the evolutionary behavior of weak discontinuities.     

Integrating Scalar Ordinary Differential Equations with Symmetry Revisited

The process of integrating an nth-order scalar ordinary differential equation with symmetry is revisited in terms of Pfaffian systems. This formulation immediately provides a completely algebraic method to determine the initial conditions and the corresponding solutions which are invariant under a one parameter subgroup of a symmetry group. To determine the noninvariant solutions the problem splits into three cases. If the dimension of the symmetry groups is less than the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions can be found by integrating a quotient Pfaffian system on a quotient space, and integrating an equation of fundamental Lie type associated with the symmetry group. If the dimension of the symmetry group is equal to the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions are obtained either by solving an equation of fundamental Lie type associated with the symmetry group, or the solutions are invariant under a one-parameter subgroup. If the dimension of the symmetry group is greater than the order of the equation, then there exists an open dense set of initial conditions where the solutions can either be determined by solving an equation of fundamental Lie type for a solvable Lie group, or are invariant. In each case the initial conditions, the quotient Pfaffian system, and the equation of Lie type are all determined algebraically. Examples of scalar ordinary differential equations and a Pfaffian system are given.     

Exact solutions of the time fractional nonlinear Schrödinger equation with two different methods

In the present paper, exact solutions of fractional nonlinear Schrödinger equations have been derived by using two methods: Lie group analysis and invariant subspace method via Riemann‐Liouvill derivative. In the sense of Lie point symmetry analysis method, all of the symmetries of the Schrödinger equations are obtained, and these operators are applied to find corresponding solutions. In one case, we show that Schrödinger equation can be reduced to an equation that is related to the Erdelyi‐Kober functional derivative. The invariant subspace method for constructing exact solutions is presented for considered equations.     

Lie group classification and invariant solutions of mKdV equation with time-dependent coefficients

This paper studies the modified Korteweg–de Vries equation with time variable coefficients of the damping and dispersion using Lie symmetry methods. We carry out Lie group classification with respect to the time-dependent coefficients. Lie point symmetries admitted by the mKdV equation for various forms for the time variable coefficients are obtained. The optimal system of one-dimensional subalgebras of the Lie symmetry algebras are determined. These are then used to determine exact group-invariant solutions, including soliton solutions, and symmetry reductions for some special forms of the equations.     

Group analysis,exact solutions and conservation laws of a generalized fifth order KdV equation

We study the generalized fifth order KdV equation using group methods and conservation laws. All of the geometric vector fields of the special fifth order KdV equation are presented. By using the nonclassical Lie group method, it is show that this equation does not admit nonclassical type symmetries. Then, on the basis of the optimal system, the symmetry reductions and exact solutions to this equation are constructed. For some special cases, we obtain additional nontrivial conservation laws and scaling symmetries.     

mKdV方程的对称与群不变解

主要考虑mKdV方程的一些简单对称及其构成的李代数,并利用对称约化的方法将mKdV方程化为常微分方程,从而得到该方程的群不变解,这是对该方程群不变解的进一步扩展.     

Symmetry and integrable canonical flows

Following the method of a group theoretic formulation of rigid body dynamics, we construct an elementary proof that f commuting generators of symmetries of an f degree-of-freedom Hamiltonian system yield integrability of the dynamics in the form of f independent translations in phase space. The integrability of the dynamical system follows directly from the trivial integrability of a particular set of group parameter velocities that are nonintegrable in the absence of symmetry, and does not rely at all upon any assumption of separability of the Hamiltonian-Jacobi partial differential equation. Our method relies upon Hamel's explanation of when one can and cannot choose group parameters as generalized coordinates, and uses the Poisson bracket formulation of mechanics that is familiar to physicists.

We formally extend Euler's theorem on rigid body motions to other transformation groups for Hamiltonian flows in phase space, and also note the analogy between nonholonomic coordinates in classical mechanics and uncertainty principles in quantum mechanics.     

Hidden symmetries and nonlocal group generators for ordinary differential equations

Hidden symmetries of ordinary differential equations (ODEs)are studied with nonlocal group generators. General forms aregiven for an exponential nonlocal group generator of an ODEthat is reduced from a higher-order ODE, which is expressedin canonical variables and which is invariant under a two-parameterLie group. The nonlocal group generator identifies a type Ihidden symmetry. Type II hidden symmetries are found in somereduction pathways of an ODE invariant under a solvable, nonabelian,three-parameter Lie group. The algorithm for the appearanceof the type II hidden symmetry is stated. General forms forthe reduced nonlocal group generator, which identifies the typeII hidden symmetry, are presented when the other two commutingoriginal group generators are in normal form.