不可压缩Navier-Stokes方程最优动力系统建模和分析

研究了同时满足任意速度边界条件和速度不可压条件的Navier-Stokes方程最优动力系统的建模方法.通过对方柱绕流问题的最优动力系统的建模与分析,发现该最优动力系统的动力学特性为极限环.同时,该最优动力系统仅使用了三个最优基函数就很好地描述了所有主要的流场特征和该问题的动力学特性,故满足任意速度边界条件和速度不可压条件Navier-Stokes方程最优动力系统的建模方法,能够用最少的基函数最大限度地描述复杂流体问题及其动力学特性.     

RBF-PU方法求解二维非局部扩散问题和近场动力学问题

采用单位分解径向基函数(radial basis function partition of unity,RBF-PU)方法,数值求解了二维非局部扩散问题和近场动力学问题。主要思想是对求解区域进行局部划分,在局部子区域上分别进行函数逼近,然后加权得到未知函数的全局逼近。这种基于方程强形式的径向基函数方法在求解非局部问题时,不需要处理网格与球形邻域求交的问题,避免了额外的一层积分计算,实施简便,计算量小。数值实验显示计算结果与解析解吻合较好,RBF-PU方法可以准确有效地求解非局部扩散方程和近场动力学方程。     

辛Runge-Kutta方法在卫星交会对接中的非线性动力学应用研究

卫星交会对接问题是实现太空平台等空间系统的关键问题之一.考虑了由于地球引力作用而引起的卫星交会对接中的非线性动力学问题.首先,采用能量方法给出Lagrange函数;然后,通过引入广义坐标和广义动量,以及Legendre变换,得到Hamilton方程;随后,采用辛Runge-Kutta方法求解该Hamilton方程,并与传统的四阶Runge-Kutta方法对比.数值结果表明:辛Runge-Kutta方法能够在积分过程中长时间保持系统的固有特性,为天体动力学问题的研究提供了良好的数值方法.     

函数与方程思想在三角问题中的应用

函数思想,是指用函数的概念和性质去分析问题、转化问题和解决问题.方程思想,是从问题的数量关系入手,将问题中的条件转化为数学模型:方程、不等式或方程与不等式的混合组,然后通过解方程(组)或不等式(组)来使问题获解.函数与方程犹如亲兄弟,彼此身上存在对方的影子,两者互相转化接轨,形成了函数与方程思想.本文将用函数与方程思想来解决三角函数的证明求值问题.     

刚塑性材料塑性动力学问题中的一般方程和通解

本文是文[1～2]的继续。本文讨论了塑性流动理论中的理想刚塑性材料的动力学问题。在引入Dirac-Pauli表象的复变函数理论后,我们可以得到用流函数和理论比例系数表示的一组(两个)所谓"一般方程"。本文还证明了塑性动力学问题的时间发展方程既非耗散型的,又非弥散型的,而其本征方程却是以应力增量的偏张量为本征函数,以理论比例系数为本征值的定态Schr?dinger方程。于是,我们使非线性塑性动力学问题成为线性定态Schr?dinger方程的求解,由此可以得到刚塑性材料塑性动力学问题的通解。     

重方法，求效益：活用函数与方程思想

函数与方程思想是四大数学思想之一,也是高考中的重要考点之一.在解决一些非函数与方程问题时,借助函数或方程的转化,将不等式、数列、三角函数、平面向量、解析几何与立体几何等相关问题转化为对应的函数或方程问题,实现化归与转化,进而利用函数或方程来分析与求解,引领并指导复习备考.     

含压力基Navier-Stokes方程最优动力系统建模和分析

研究了采用压力基函数和速度基函数的Navier-Stokes方程的最优截断低维动力系统建模理论.在黏性不可压缩流体中模拟了并排三方柱绕流流场,对此流场进行了含压力基函数和速度基函数的Navier-Stokes方程的最优动力系统建模,并以此为工具分析了三方柱绕流最优动力系统的动力学特性.该研究得到了如下结论:三方柱绕流的最优动力系统的动力学行为为混沌,它与双方柱绕流场的极限环动力学特性有着本质的区别,因此可以通过多柱绕流增进尾流的复杂性,从而促进流体混合.     

几种广义非线性发展方程的新解

本文研究了构造了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解的问题.利用三种辅助方程及其新解,获得了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解.这些解由双曲余割函数、双曲正切函数、双曲正割函数、双曲余切函数和余割函数组成.     

函数与方程问题中的转化

<正>函数的零点与方程根的问题是高中数学的重要内容,也是高考热点考题之一.往往涉及的函数与方程都比较复杂,并不是能直接解出零点或能求出方程根的问题,它需要将复杂的函数或方程问题转化为我们熟悉的函数或方程问题,并结合不同函数图象的位置关系达到求解的目的.     

圆柱壳的轴对称平面应变弹性动力学解

给出一种圆柱壳的轴对称平面应变弹性动力学问题的解析方法。首先通过引入一特定函数将非齐次边界条件化为齐次边界条件,然后利用分离变量法将位移减去特定函数的量展开为关于贝塞尔函数和时间函数乘积的级数,并由贝塞尔函数的正交性,导出时间函数的方程,容易求得此方程的解。将两者叠加可得弹性动力学问题的位移解。运用此方法,可以避免积分变换,并适宜于各种载荷。文中给出了各向同性和柱面各向同性圆柱壳内表面和实心圆柱外表面受冲击荷载作用以及内表面固定的柱面各向同性圆柱壳外表面受冲击荷载作用的数值结果。     

Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.     

Mathematical aspects of the Maxwell problem

We study the large-time behaviour of global smooth solutions to the Cauchy problem for hyperbolic regularization of conservation laws. An attracting manifold of special smooth global solutions is determined by the Chapman projection onto the phase space of consolidated variables. For small initial data we construct the Chapman projection and describe its properties in the case of the Cauchy problem for moment approximations of kinetic equations. The existence conditions for the Chapman projection are expressed in terms of the solvability of the Riccati matrix equations with parameter.     

Navier-Stokes approximation and problems of the Chapman-Enskog projection for kinetic equations

The purpose of this paper is to investigate problems of the Navier-Stokes approximation to kinetic equations in terms of the so-called Chapman-Enskog projection. One considers properties of the Chapman-Enskog projection for the Cauchy problem for moment approximations of the kinetic equation and primarily the Chapman-Enskog projection for the Boltzmann-Peierls kinetic equation. The existence of the Chapman-Enskog projection for the Cauchy problem is proved for the phase space of conservative variables (phenomena of nonlinear diffusion) and for the phase space of physical variables (the second sound projection). __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 184–225, 2005.     

退化抛物-双曲方程动力学解的唯一性

主要研究系数显含有时间和空间变量的退化抛物-双曲型方程柯西问题动力学解的唯一性.首先推广了这种类型方程的动力学公式,在给定系数适当的光滑性条件下,得到了动力学解的唯一性.     

The existence of global solutions to the Cauchy problem for discrete kinetic equations. II

We continue the study of the global solvability of the Cauchy problem for discrete kinetic equations and consider the general case of complex data of the problem. We prove the existence of a global solution and obtain its representation. Bibliography: 10 titles.     

Solvability of matrix Riccati equations

The paper is aimed at studying solvability conditions for the quadratic matrix Riccati equation that arises in connection with the Chapman–Enskog projection for the Cauchy problem and the mixed problem for moment approximations of kinetic equations. The structure of the matrix equation allows for the formulation of necessary and sufficient conditions for the existence of solutions in terms of eigenvectors and associated vectors of the coefficient matrix.     

Solvability of quadratic matrix equations

Solvability conditions are studied in this paper for a quadratic matrix Riccati equation arising in studies of the Chapman-Enskog projection for a Cauchy problem and a mixed problem for momentum approximations of kinetic equations. The structure of the matrix equation permits one to formulate necessary and sufficient solvability conditions in terms of eigenvectors and associated vectors for the matrix composed from the coefficients.     

Exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term

The present work derives the exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term. The work also emphasizes the role of reaction–diffusion models as important particular cases of much more general equations in the kinetic theory of active particles. The analytical expression derived shows the structure of the solution and the contributions of different terms of the model to it. The result obtained enables one to solve the Cauchy problem indicated by using the exact analytical representation rather than numerical methods, which are usually time-consuming, especially when the number of spatial dimensions is greater than 2.     

Differentiability of solutions of some evolution equations in nonreflexive Banach spaces

Summary The purpose of this paper is to present a regularity result that provides a unified treatment of the Cauchy problem for certain nonlinear partial differential equations that appear in kinetic theory. Part. 1 contain the main theorem, based on the theory of evolution equations. In part 2 it is indicated how these abstract results are applicable to the spatially homogeneous Boltzmann equazion and to the kinetic equation of vehicular traffic. Entrata in Redazione il 28 giugno 1978. Work performed under the auspices of the G.N.A.F.A. of the National Research Couucil.     

On the Maxwell problem

We study the large-time behavior of global smooth solutions to the Cauchy problem for hyperbolic regularization of conservation laws. An attracting manifold of special smooth global solutions is determined by the Chapman–Enskog projection onto the phase space of consolidated variables. For small initial data we construct the Chapman–Enskog projection and describe its properties in the case of the Cauchy problem for moment approximations of kinetic equations. The existence conditions for the Chapman–Enskog projection are expressed in terms of the solvability of the Riccati matrix equations with parameter. Bibliography: 21 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 27–63.