Symmetries of Linear and Linearizable Systems of Differential Equations

Symmetries (point, classical, and higher) of linear and linearizable differential equations and systems are studied, including the cases of ordinary equations, equations of the Burgers type, and Lax pairs.     

Quantum and Integral (Co)Bordisms in Partial Differential Equations

Characterizations of quantum bordisms and integral bordisms in PDEs by means of subgroups of usual bordism groups are given. More precisely, it is proved that integral bordism groups can be expressed as extensions of quantum bordism groups and these last are extensions of subgroups of usual bordism groups. Furthermore, a complete cohomological characterization of integral bordism and quantum bordism is given. Applications to particular important classes of PDEs are considered. Finally, we give a complete characterization of integral and quantum singular bordisms by means of some suitable characteristic numbers. Some examples of interesting PDEs which arise in physics are also considered where existence of solutions with change of sectional topology (tunnel effect) is proved. As an application, we relate integral bordism to the spectral term that represents the space of conservation laws for PDEs. This also gives a general method to associate in a natural way a Hopf algebra to any PDE.     

Comparison of Approximate Symmetry Methods for Differential Equations

Two current approximate symmetry methods and a modified new one are contrasted. Approximate symmetries of potential Burgers equation and non-Newtonian creeping flow equations are calculated using different methods. Approximate solutions corresponding to the approximate symmetries are derived for each method. Symmetries and solutions are compared and advantages and disadvantages of each method are discussed in detail.     

Essential Conservation Laws for Equations with Infinite Symmetries

We consider partial differential equations of variational problems with infinite symmetry groups. We study local conservation laws associated with arbitrary functions of one variable in the group generators. We show that only symmetries with arbitrary functions of dependent variables lead to an infinite number of conservation laws. We also calculate local conservation laws for the potential Zabolotskaya-Khokhlov equation for one of its infinite subgroups.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 190–198, July, 2005.     

An Example of (3+1)-Dimensional Integrable System

In this paper an example of the (3+1)-dimensional integrable system is considered and the infinite series of divergent forms are described. The classical symmetries for this system, the factor-equation for these symmetries, and exact solutions of this system are found.     

微分方程递归算子的一种推广

微分方程的多数重要的递归算子都是积微分算子。在试图获得整个对称族时人们常常遇到困难。有时甚至由于缺乏精确性而导致伪对称。本文给出递归算子一种推广,其在某种程度上消除了这些问题。文中还给出了若干重要例子说明这种推广及其重要性。     

Stochastic Partial Differential Equations on Manifolds~II. Nonlinear Filtering

The conditional law of an unobservable component x(t) of a diffusion (x(t),y(t)) given the observations {y(s):s[0,t]} is investigated when x(t) lives on a submanifold of . The existence of the conditional density with respect to a given measure on is shown under fairly general conditions, and the analytical properties of this density are characterized in terms of the Sobolev spaces used in the first part of this series.     

二阶非线性泛函微分方程的振动性质

研究了一类二阶非线性泛函微分方程解的振动性质.在一定条件下,应用分类讨论和分析的方法,建立了一个新的振动性定理,推广和改进了已知的结果.     

On the Geometry of Point-Transformation Invariant Class of Third-Order Ordinary Differential Equations

Applying geometric methods, we study a class of third-order ordinary differential equations closed with respect to point transformations. We associate with such an equation the pseudovector fields formed by its coefficients. The equation possesses a maximal algebra of point-transformation symmetries if five pseudovector fields vanish.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 719–726.Original Russian Text Copyright ©2005 by V. V. Kartak.     

Stability Analysis of Runge-Kutta Methods for Non-Linear Delay Differential Equations

This paper is concerned with the numerical solution of delay differential equations(DDEs). We focus on the stability behaviour of Runge-Kutta methods for nonlinear DDEs. The new concepts of GR(l)-stability, GAR(l)-stability and weak GAR(l)-stability are further introduced. We investigate these stability properties for (k, l)-algebraically stable Runge-Kutta methods with a piecewise constant or linear interpolation procedure.     

Local conservation laws,symmetries, and exact solutions for a Kudryashov‐Sinelshchikov equation

In this paper, we consider a Kudryashov‐Sinelshchikov equation that describes pressure waves in a mixture of a liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer between liquid and gas bubbles. We show that this equation is rich in conservation laws. These conservation laws have been found by using the direct method of the multipliers. We apply the Lie group method to derive the symmetries of this equation. Then, by using the optimal system of 1‐dimensional subalgebras we reduce the equation to ordinary differential equations. Finally, some exact wave solutions are obtained by applying the simplest equation method.     

一类二阶非线性泛函微分方程的振动性定理

研究了一类较为广泛的二阶非线性泛函微分方程解的振动性质.在一定条件下,利用广义黎卡提变换、积分平均技巧和分类讨论的方法建立了两个新的振动准则,改进和推广了已知的一些结果.     

一类二阶非线性泛函微分方程的振动性质

研究了一类较为广泛的二阶非线性泛函微分方程解的振动性质.在一定条件下,利用广义黎卡提变换和积分平均技巧建立了方程(1)的两个新的振动准则,推广和改进了已知的一些结果.     

几类非线性微分方程闭轨线的不存在性

本文研究几类非线性微分方程不存在闭轨线的条件,所得结果可用于判别方程■ f_1(x)■ f_2(x)■ g(x)=0和■ f(x,■)■ g(x)=0闭轨线的不存在性.     

Nonlinear Stability of General Linear Methods for Delay Differential Equations

This paper is devoted to a study of nonlinear stability of general linear methods for the numerical solution of delay differential equations in Hilbert spaces. New stability concepts are further introduced. The stability properties of (k,p,q)-algebraically stable general linear methods with piecewise constant or linear interpolation procedure are investigated. We also discuss stability of linear multistep methods viewed as a special subset of the class of general linear methods.     

线性常微分方程(组)的算子方法介绍及其研究展望

综述了线性微分方程(组)的算子方法,侧重地介绍了作者所发展的一系列方法和重要的结果与解公式.提出了算子方法研究的几点展望.     

带有阻尼项的二阶非线性微分方程的振动性

利用一类Φ(t,s,l)型的新函数,对带有阻尼项的二阶非线性微分方程建立了新的振动准则,推广和改进了已有的一些振动结果.     

二阶非线性阻尼脉冲时滞微分方程解的振动准则

利用了Lakshmikantham等人建立的脉冲微分不等式讨论了一类二阶非线性脉冲微分方程解的振动性质,获得了此类方程振动所应具备的充分条件,同时改进了一些已知的结果,最后用一个具体的例子说明了是否带有脉冲对微分方程的振动性有很大的影响.     

一类二阶非线性脉冲微分方程解的振动性

讨论了一类二阶非线性脉冲微分方程解的振动性质,并得到了这类方程解的振动的一组充分条件.     

二阶非线性泛函微分方程的渐近性质

研究了一类二阶非线性泛函微分方程非振动解的渐近性质及其分类.在一定条件下,建立了三个新的渐近性定理,推广和改进了已知的结果.