Fokker Planck方程的非古典广义势对称及精确解

分析了Fokker-Planck方程的非古典势对称,通过广义势系统而不是一般势系统求得了这些非古典势对称.文中得到了这些方程的新的对称,同时也得到了伴随系统的新的对称,并用其求出了一些精确解.这些解对进一步研究Fokker-Planck方程所描述的物理现象具有广泛的应用价值.     

Fokker-Planck方程的非古典势对称群及新显式解

本文利用一种新方法对Fokker- Planck方程的非古典势对称群生成元进行研究,找到方程的几个非古典势对称群生成元,并采用非古典对称群方法由这些对称群生成元构造得到Fokker- Planck方程的相应显式解.这些新显式解不能由Fokker -Planck方程本身的Lie对称或Li-e B cklund对称来获得.在验证所求得显式解的过程中,还发现并得到了另外几个显式解.这些新显式解则不能由Fokker -Planck方程本身的Lie对称,Lie- B cklund对称或非古典势对称来获得.文章表明,通过偏微分方程的非古典势对称群生成元来寻找其显式解是可能的.     

几个非线性偏微分方程的非古典对称及相似解

研究了一些非线性偏微分方程的非古典势对称和非古典对称,得到了某些方程的新的势对称和新的对称,同时也得到了其伴随系统的新的对称,并求出了一些相似解．这些解对进一步研究这些非线性偏微分方程所描述的物理现象具有广泛的应用价值．     

一类广义Riccati矩阵方程对称解的双迭代算法

研究了一类广义系统控制理论导出的Riccati矩阵方程对称解的数值计算方法．运用牛顿算法将Riccati矩阵方程的对称解问题转化为线性矩阵方程的对称解或者对称最小二乘解问题,采用修正共轭梯度法解决导出的线性矩阵方程的对称解问题,可建立求Riccati矩阵方程对称解的双迭代算法．数值算例表明,双迭代算法是有效的．     

求矩阵方程AXB=C的双对称最小二乘解的迭代算法

基于求解线性代数方程组的共轭梯度法的思想,通过特殊的变形与近似处理,建立了求矩阵方程AXB=C的双对称最小二乘解的迭代算法,并证明了迭代算法的收敛性.不考虑舍入误差时,迭代算法能够在有限步计算之后得到矩阵方程的双对称最小二乘解;选取特殊的初始矩阵时,还能够求得矩阵方程的极小范数双对称最小二乘解.同时,也能够给出指定矩阵的最佳逼近双对称矩阵.算例表明,迭代算法是有效的.     

一类强非线性系统共振周期解的渐近分析

强非线性系统经引入参数变换,并在一定的假设条件下,可转化为弱非线性系统．将其解展成为改进的傅立叶级数后,利用参数待定法可方便地求出强非线性系统的共振周期解．研究了Duffing方程的主共振、Van der Pol方程的3次超谐共振和Van der Pol-Mathieu方程的1／2亚谐共振周期解．这些例子表明近似解与数值解非常吻合。     

关于一般耦合矩阵方程的迭代对称解

本文构造了一个有效的迭代方法(CGL)去求解一般耦合矩阵方程的对称解.若一般耦合矩阵方程关于对称解相容,则对于任意给定的初始对称矩阵组,利用所构造的迭代算法,都能在有限步迭代出所求问题的一组对称解,若选用一些特殊的初值,则可获得矩阵方程的极小范数对称解.最后的数值例子表明了所给算法的有效性.     

线性矩阵方程异类约束最小二乘解的迭代算法

多矩阵变量线性矩阵方程(LME)约束解的计算问题在参数识别、结构设计、振动理论、自动控制理论等领域都有广泛应用。本文借鉴求线性矩阵方程(LME)同类约束最小二乘解的迭代算法,通过构造等价的线性矩阵方程组,建立了求多矩阵变量LME的一种异类约束最小二乘解的迭代算法,并证明了该算法的收敛性。在不考虑舍入误差的情况下,利用该算法不仅可在有限步计算后得到LME的一组异类约束最小二乘解,而且选取特殊初始矩阵时,可求得LME的极小范数异类约束最小二乘解。另外,还可求得指定矩阵在该LME的异类约束最小二乘解集合中的最佳逼近解。算例表明,该算法是有效的。     

多变量矩阵方程组一种异类约束解的迭代算法

借鉴求线性矩阵方程组同类约束解的修正共轭梯度法,建立了求多个未知矩阵的线性矩阵方程组的一种异类约束解的修正共轭梯度法,并证明了该算法的收敛性．利用该算法不仅可以判断矩阵方程组的异类约束解是否存在,而且在有异类约束解,且不考虑舍入误差时,可在有限步计算后求得矩阵方程组的一组异类约束解;选取特殊初始矩阵时,可求得矩阵方程组的极小范数异类约束解．另外,还可求得指定矩阵在该矩阵方程组异类约束解集合中的最佳逼近．算例表明,该算法是有效的．     

一类离散时间代数Riccati矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在线性二次优化问题中遇到的含未知矩阵之逆的离散时间代数Riccati矩阵方程(DTARME)转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求DTARME的对称解的双迭代算法。双迭代算法仅要求DTARME有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定。数值算例表明双迭代算法是有效的。     

Invariant solutions and conservation laws to nonconservative FP equation

We generate conservation laws for the one dimensional nonconservative Fokker–Planck (FP) equation, also known as the Kolmogorov forward equation, which describes the time evolution of the probability density function of position and velocity of a particle, and associate these, where possible, with Lie symmetry group generators. We determine the conserved vectors by a composite variational principle and then check if the condition for which symmetries associate with the conservation law is satisfied. As the Fokker–Planck equation is evolution type, no recourse to a Lagrangian formulation is made. Moreover, we obtain invariant solutions for the FP equation via potential symmetries.     

对称在破裂孤立子方程中的应用

首先对带有积分项的破裂孤立子方程（breaking soliton equation）进行变换,然后利用待定系数法求出它的对称,通过验证知道原方程的李群能构成李代数,再利用优化系统对原方程进行约化,求出了原方程的一些新解。     

Unified detection of skewed rotation,reflection and translation symmetries from affine invariant contour features

Symmetry detection is significant for object detection and recognition since symmetries are salient cues for distinguishing geometrical structures from cluttered backgrounds. This paper proposes a unified framework to detect rotation, reflection and translation symmetries simultaneously on unsegmented real-world images. The detection is performed based on affine invariant contour features, and is applicable under skewed imaging with distortions. Contours on a natural image are first matched to each other to find affine invariant matching pairs, which are then classified into three groups using a sign change criterion corresponding to the three types of symmetries. The three groups are used to vote for the corresponding symmetries: the voting for rotation utilizes a scheme of short line segments; the voting for reflection is based on a parameterization of axis equation, and the voting for translation uses a cascade-like approach. Experimental results of real-world images are provided with quantitative evaluations, validating that the proposed framework achieves desired performance.     

Application of He's homotopy perturbation and He's variational iteration methods for solution of Benney–Lin equation

In this paper, we study the Benney–Lin equation using He's homotopy perturbation method (HPM) and He's variational iteration method (VIM). We compare HPM and VIM methods and show that the results of the HPM method are in excellent agreement with the results of the VIM method and the obtained solutions are shown graphically. Several cases are considered to apply HPM and VIM, which demonstrate the reliability and efficiency of these two methods in solving such a complicated equation with various initial conditions.     

An Algorithm to Determine Potential Systems in Mathematica

A symmetry analysis of differential equations plays an important role in discovering new solutions. In this article potential symmetries characterized by nonlocal transformations are introduced and an algorithm implemented in the computer algebra system Mathematica is presented which determines automatically potential systems and the corresponding potential symmetries. The possibilities of this algorithm are discussed by the examples of a nonlinear telegraph equation and the axial symmetric wave equation.     

Group properties of crossover and mutation

It is supposed that the finite search space omega has certain symmetries that can be described in terms of a group of permutations acting upon it. If crossover and mutation respect these symmetries, then these operators can be described in terms of a mixing matrix and a group of permutation matrices. Conditions under which certain subsets of omega are invariant under crossover are investigated, leading to a generalization of the term schema. Finally, it is sometimes possible for the group acting on omega to induce a group structure on omega itself.     

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.