对称分类在非线性偏微分方程组边值问题中的应用

研究了微分方程对称分类在非线性偏微分方程组边值问题中的应用.首先,利用偏微分方程(组)完全对称分类微分特征列集算法确定了给定非线性偏微分方程组边值问题的完全对称分类;其次,利用一个扩充对称将非线性偏微分方程组边值问题约化为常微分方程组初值问题;最后,利用龙格-库塔法求解了常微分方程组初值问题的数值解.     

扰动扩散方程初值问题的近似对称约化

本文利用近似广义条件对称方法研究一类带有源项的非线性扩散方程的初值问题.给出所研究方程的分类并将偏微分方程的初值问题约化为常微分方程的初值问题,通过求解约化后的常微分方程组可得相对应偏微分方程初值问题的近似解.     

Local Well-Posedness for Membranes in the Light Cone Gauge

In this paper we consider the classical initial value problem for the bosonic membrane in light cone gauge. A Hamiltonian reduction gives a system with one constraint, the area preserving constraint. The Hamiltonian evolution equations corresponding to this system, however, fail to be hyperbolic. Making use of the area preserving constraint, an equivalent system of evolution equations is found, which is hyperbolic and has a well-posed initial value problem. We are thus able to solve the initial value problem for the Hamiltonian evolution equations by means of this equivalent system. We furthermore obtain a blowup criterion for the membrane evolution equations, and show, making use of the constraint, that one may achieve improved regularity estimates.     

变系数瞬态热传导问题边界元格式的特征正交分解降阶方法

提出了一种基于边界元法求解变系数瞬态热传导问题的特征正交分解(POD)降阶方法,重组并推导出变系数瞬态热传导问题适合降阶的边界元离散积分方程,建立了变系数瞬态热传导问题边界元格式的POD降阶模型,并用常数边界条件下建立的瞬态热传导问题的POD降阶模态,对光滑时变边界条件瞬态热传导问题进行降阶分析.首先,对一个变系数瞬态热传导问题,建立其边界域积分方程,并将域积分转换成边界积分;其次,离散并重组积分方程,获得可用于降阶分析的矩阵形式的时间微分方程组;最后,用POD模态矩阵对该时间微分方程组进行降阶处理,建立降阶模型并对其求解.数值算例验证了本文方法的正确性和有效性.研究表明:1)常数边界条件下建立的低阶POD模态矩阵,能够用来准确预测复杂光滑时变边界条件下的温度场结果;2)低阶模型的建立,解决了边界元法中采用时间差分推进技术求解大型时间微分方程组时求解速度慢、算法稳定性差的问题.     

一种求解逆风下火焰沿柱状燃料传播的数值方法

1引言近十几年来，国内外许多人对火焰沿薄燃料层的传播进行了研究，但很少涉及到火焰沿柱状燃料的传播。图1为所研究问题的物理模型。火焰的传播涉及到气固两相的相互作用，两相界面上存在能量和质量的交换，气相中存在着气气间的化学反应，固相中存在着裂解反应。当参考坐标系团结在火焰前锋上时，该火焰传播问题成为一准稳态问题，燃料以一固定速度供给火焰。2数学模型2．1守恒方程气相的所有守恒方程均服从下列通用方程：其中、，L和一分别表示通用变量、广义扩散系数和广义源项。固相包括：质量守恒方程：。。能量守恒方程：2．2边界…     

Interactions of an elastic solid with a viscous fluid: Eigenmode analysis

In this paper we study the interaction of a viscous fluid with an elastic solid. Of particular interest are the eigenmodes of the coupled system. Starting from the Navier-Stokes equations for the fluid and the linear elasticity equations for the solid, we derive the linear equations governing the motion of the system. It is shown how a variational formulation of the problem may be obtained by re-scaling the displacement unknowns. The finite-element technique is then used to discretize the equations. The resulting quadratic eigenvalue problem is solved by means of an inverse iteration procedure.     

蒙特卡罗方法在解微分方程边值问题中的应用

介绍了蒙特卡罗方法的基本原理以及随机数的产生方法。基于蒙特卡罗方法的思想,结合有限差分方法,建立了求解微分方程边值问题的随机概率模型,并以第一类边界条件的拉普拉斯方程和一个给定初值及边界条件的非稳态热传导方程为数值算例,研究了蒙特卡罗方法在求解微分方程边值问题中的应用。结果表明:利用蒙特卡罗方法,不仅可以有效解决给定边界条件的微分方程,对于给定初值条件的微分方程,也可以从时域有限差分方程出发,采用蒙特卡罗方法进行求解。数值模拟和对误差的理论分析均表明,增加蒙特卡罗试验中的模拟粒子点数,可以提高计算结果的精度。     

EQUIVALENCE CLASSES OF THE SECOND ORDER ODEs WITH THE CONSTANT CARTAN INVARIANT

Second order ordinary differential equations that possesses the constant invariant are investigated. Four basic types of these equations were found. For every type the complete list of nonequivalent equations is issued. As the examples the equivalence problem for the Painleve II equation, Painleve III equation with three zero parameters, Emden equations and for some other equations is solved.     

Resolution of the Navier-Stokes equations in vorticity-velocity formulation via control techniques

The velocity-vorticity formulation of the Navier-Stokes equations is turned into an optimal control problem subject to linear constraints which consist of Laplacian equations for the velocity and for the vorticity. Various optimization algorithms can be used for the numerical resolution of the optimal control problem in question; in particular, the use of a conjugate-gradient method is discussed.     

Uniqueness of Solutions in Thermopiezoelectricity of Nonsimple Materials

This article presents the theory of thermopiezoelectricity in which the second displacement gradient and the second electric potential gradient are included in the set of independent constitutive variables. This is achieved by using the entropy production inequality proposed by Green and Laws. At first, appropriate thermodynamic restrictions and constitutive equations are obtained, using the well-established Coleman and Noll procedure. Then, the balance equations and the constitutive equations of linear theory are derived, and the mixed initial-boundary value problem is set. For this problem a uniqueness result is established. Next, the basic equations for the isotropic case are derived. Finally, a set of inequalities is obtained for the constant constitutive coefficients of the isotropic case that, on the basis on the previous theorem, ensure the uniqueness of the solution of the mixed initial-boundary value problem.     

A Hierarchy of Integrable Nonlinear Lattice Equations and New Integrable Symplectic Map

A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related tothis spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are givenby nonlinearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resultingintegrable lattice equations.     

Phase measurement of waves that obey nonlinear equations

We consider the problem of phase retrieval for classical and quantum wave fields that obey a wide class of nonlinear wave equations. This problem is tackled by means of a suitable generalization of existing methods based on the linear transport-of-intensity equation. The extension of these ideas to systems of coupled nonlinear wave equations is also given.     

Yang's Gravitational Theory

Yang's pure space equations generalize Einstein's gravitational equations, while coming from gauge theory. We study these equations from a number of vantage points: summarizing the work done previously, comparing them with the Einstein equations and investigating their properties. In particular, the initial value problem is discussed and a number of results are presented for these equations with common energy-momentum tensors.     

A mixed problem for the Einstein equations and an approximate method of solution

The present paper poses a mixed problem for the Einstein equations. A combined method for solving the problem is introduced. The method consists of a combination of the finite-difference method for the time coordinate, and Galerkin's method for solving the system of equations so obtained. Existence and uniqueness conditions are found for the mixed problem in an appropriately introduced functional space. The convergence conditions for the method are found.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 113–118, May, 1976.     

The Geometry of the Dirac Equation

We discuss the problem of the derivation and the interpretation of metric tensors and generalized equations of motion for test particles from quasilinear spinor equations.     

Birkhoff系统的离散最优控制及其在航天器交会对接中的应用

由于非线性,最优控制问题通常依赖于数值求解,即通过离散目标泛函和受控运动方程转化为一有限维的非线性最优化问题.最优控制问题中的受控运动方程在表示为受控Birkhoff方程的形式之后,可以利用受控Birkhoff方程的离散变分差分格式进行离散.与按照传统差分格式近似受控运动方程相比,此途径可以诱导更加真实可靠的非线性最优化问题,进而也会诱导更加精确有效的离散最优控制.应用于航天器交会对接问题,该种数值求解最优控制问题的方法在较大时间步长的情况下仍然求得了一个有效实现交会对接的离散最优控制.模拟结果验证了该方法的有效性.     

Initial-value problem for plasma oscillations in a uniform magnetic field

The solution is given of the initial-value problem for the nonrelativistic linearised Vlasov-Maxwell equations describing longitudinal and transverse plasma oscillations in an external uniform magnetic field. The problem is solved for all directions of propagation except normal to the external magnetic field, and the equilibrium distribution is not assumed isotropic. The method of solution is an extension of Van Kampen's eigenfunction expansion technique, already developed considerably by Zelazny and McCure, in which the problem is reduced to the solution of a system of singular integral equations.     

A new type of disconnectedness problem in a field-theory model of the NNπ system

When treated as an effective three-body problem in the framework of a simple field-theory model, the NNπ system acquires, in addition to the disconnected subsystem interactions usually considered, a new type of disconnected driving term, possible only for non-conserved particles such as the π. These terms pose a disconnectedness problem more intricate than that solved by Faddeev's equations or their known modifications for connected three-body forces. The solution of this problem in terms of a set of connected-kernel integral equations is presented.     

NON-LINEAR TRANSVERSE VIBRATIONS OF A SIMPLY SUPPORTED BEAM CARRYING CONCENTRATED MASSES

An Euler-Bernoulli beam carrying concentrated masses is considered to be a beam-mass system. The beam is simply supported at both ends. The non-linear equations of motion are derived including stretching due to immovable end conditions. The stretching introduces cubic non-linearities into the equations. Forcing and damping terms are also included. Exact solutions for the natural frequencies are given for the linear problem. For the non-linear problem, an approximate solution using a perturbation method is searched. Non-linear terms of the perturbation series appear as corrections to the linear problem. Amplitude and phase modulation equations are obtained. Non-linear free and forced vibrations are investigated in detail. The effect of the positions, magnitudes and number of the masses are investigated.     

Post-Newtonian equations for the metric perturbation generated by a rotating elastic earth

By using Synge's approximation method to describe the unperturbed problem, we obtain the equations for the gravitational field perturbation and the Lagrangian displacement that occur when an isolated and initially self-gravitating spherical and static elastic earth gets into steady rotation. These equations are explicitly derived in an order of approximation for the initial problem where both rotation and elastic structure manifest themselves in the perturbed state.