A Class of Sparse Spectral Operators for Inversion of Powers of the Laplacian in N Dimensions

For inversion of the Laplacian subject to Dirichlet boundary conditions and, more generally, for the kth power of the Laplacian subject to boundary conditions on the function and its first k – 1 derivatives in the normal coordinate, there is a sparse symmetric, well-conditioned, projection of the operator that results from an expansion in associated Legendre polynomials.     

Zakai equation of nonlinear filtering with unbounded coefficients. The case of dependent noises

In this paper, we prove that the unnormalized filter associated with nonlinear filtering problems with dependent noises and a one-dimensional observations process the coefficients of which are unbounded solves a parabolic stochastic partial differential equation, the Zakai equation. The robust form of the Zakai equation is also computed.     

Numerical solution of GRLW equation using Sinc-collocation method

In this paper, we present a meshfree technique for the numerical solution of the generalized regularized long wave (GRLW) equation. This approach is based on a global collocation method using Sinc basis functions. The propagation of single solitons and the interaction of two solitary waves are used to validate the method which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the method.     

求一类非线性偏微分方程解析解的简洁构造算法

通过引入一个变换,利用齐次平衡原理和选准一个待定函数来构造求解一类非线性偏微分方程解析解的算法.作为实例,我们将该算法应用到了mKdV方程,KdV-Burgers方程和KdV-Burgers-Kuramoto方程.借助符号计算软件Mathematica获得了这些方程的解析解.不难看出,该方法不仅简洁,而且有望进一步扩展.     

Soave汽液平衡模型在苯与乙烯烷基化反应动力学中的应用

实验在等温无梯度反应器内进行 ,使用 12 0～ 140mesh的FX - 0 2催化剂 ,用量 85 0～ 45 0 0mg,温度 130～2 10± 1℃ ,压力 1 95± 0 0 2MPa ,搅拌转速 10 0 0rpm下 ,研究了苯与乙烯烷基化反应动力学。在研究过程中 ,利用Soave汽液平衡模型计算了苯与乙烯烷基化反应中体系的汽液相平衡 ,从而通过计算 ,取得了动力学实验中不能直接测得的液相组成数据 ,并在此基础上进行了动力学模型的参数估值。参数估值过程采用改进的Gauss New ton法 ,以最小二乘为准则直接对常微分形式的反应动力学模型进行估值。对反应动力学模型的方差分析和残差分析的显著性检验表明 ,反应动力学模型基本无缺陷 ,其对实验数据的拟合也比较好。由此表明 ,Soave汽液平衡模型应用于苯与乙烯的烷基化反应动力学计算是成功的     

On exponential stability of linear non-autonomous functional differential equations of neutral type

General linear non-autonomous functional differential equations of neutral type are considered. A novel approach to exponential stability of neutral functional differential equations is presented. Consequently, explicit criteria are derived for exponential stability of linear non-autonomous functional differential equations of neutral type. A brief discussion to the obtained results and illustrative examples are given.     

Bounds for the eigenvalues of the solution of the unified algebraic Riccati matrix equation

In recent years, several eigenvalues bounds have been investigated separately for the solutions of the continuous and the discrete Riccati and Lyapunov matrix equations. In this paper, lower bounds for the eigenvalues of the solution of the unified Riccati equation (relatively to continuous and discrete cases), are presented. In the limiting cases, the results reduce to some new bounds for both the continuous and discrete Riccati equation.     

非线性演化方程的新Jacobi椭圆函数解

基于sinh-Gordon方程的椭圆函数解,构造新的试探解来扩展sinh-Gordon方程展开法.利用该方法研究了KdV-mKdV方程,双sine-Gordon方程和BBM方程,获得了这些方程的新Jacobi椭圆函数解.该方法也能用来求解其他数学物理中的非线性演化方程.     

The Unified Frame of Alternating Direction Method of Multipliers for Three Classes of Matrix Equations Arising in Control Theory

In this paper, the unified frame of alternating direction method of multipliers (ADMM) is proposed for solving three classes of matrix equations arising in control theory including the linear matrix equation, the generalized Sylvester matrix equation and the quadratic matrix equation. The convergence properties of ADMM and numerical results are presented. The numerical results show that ADMM tends to deliver higher quality solutions with less computing time on the tested problems.     

An analytic algorithm of Lane-Emden type equations arising in astrophysics using modified Homotopy analysis method

Lane-Emden type equation models many phenomena in mathematical physics and astrophysics. It is a nonlinear differential equation which describes the equilibrium density distribution in self-gravitating sphere of polytropic isothermal gas, has a singularity at the origin, and is of fundamental importance in the field of stellar structure, radiative cooling, modeling of clusters of galaxies. An efficient analytic algorithm is provided for Lane-Emden type equations using modified homotopy analysis method, which is different from other analytic techniques as it itself provides us with a convenient way to adjust convergence regions even without Pade technique. Some examples are given to show its validity.     

Temporal analysis of power law liquid jets

In this paper we investigate the breakup mechanisms of power law liquid jets. The viscosity of the liquid is represented the Carreau-Yasuda model, and the surface tension of the liquid jet has a variation (gradient) along the jet axial direction. The surface tension gradient may be introduced by the thermal disturbance of the jet surface as it comes of out an orifice. The Carreau-Yasuda fluid has a power law viscosity bounded by two plateaus, the higher plateau at zero strain rate, μ₀, and the lower plateau at the infinite strain rate, μ_∞. The governing equation for the surface profile of the liquid jet is derived in the forms of a partial differential equation (PDE), as well as an ordinary differential equation (ODE). The PDE and ODE are solved for various cases of Carreau-Yasuda fluid to study the effect of fluid properties on jet breakup. The effects of various parameters on the instability behavior are studied in comparison with two Newtonian jets with upper and lower bound viscosities, μ₀ and μ_∞. A number of quantitative conclusions and sensitivities on the instability behavior of non-Newtonian jets are investigated. It is found that the jet breakup mechanism depends on the properties of the fluid as well as the wave number of the thermal disturbance that causes the surface tension gradient. In contrast to the Newtonian liquid where the jet surface profile has the same frequency as the surface tension gradient, the nonlinear nature of the Carreau-Yasuda constitutive behavior may enable the jet surface profile at frequencies higher than that of the surface tension gradient. This leads to significant surface profile oscillation within one wavelength of the surface tension gradient and the generation of small satellite drops. It is worth noting that at a small wave number the breakup time for the Carreau-Yasuda fluid maybe shorter than that of the Newtonian jet with μ_∞, although the Newtonian jet has a lower viscosity.     

Matrix bounds of the discrete ARE solution

This paper provides new lower and upper matrix bounds of the solution to the discrete algebraic Riccati equation. The lower bound always works if the solution exists. The upper bounds are presented in terms of the solution of the discrete Lyapunov equation and its upper matrix bound. The upper bounds are always calculated if the solution of the Lyapunov equation exists. A numerical example shows that the new bounds are tighter than previous results in many cases.     

解析式状态方程计算含临界点方阱流体的性质

含临界区流体的物性计算一直是热力学领域的难点。一般认为，解析式状态方程不可能同时很好地计算远离临界点和靠近临界点流体的热力学性质。该文目的就是验证解析式方程在计算热力学性质特别是接近临界点附近的热力学性质适用性。通过计算机分子模拟数据，研究了各种解析式状态方程计算接近临界区的方阱流体的物性的可能性，结果表明，经过参数重新拟合新的解析式方程，可以很好地计算方阱流体的相平衡、密度等性质，直到对比温度接近临界点0．05处。临界点温度、压力和密度的计算值和分子模拟值也基本符合。该文结果可以为解析式方程计算范围的扩大提供依据，从而试图解析式在工程方面有更加广泛的应用。     

Decreasing the dimension of a system of scalar equations in the solution of diffraction problems

A vector integral equation is constructed, and an algebraic representation of the equation is considered in the solution of a boundary-value problem in an ideally conducting polyhedron. Moreover, due to the introduction of local coordinate systems on each of the faces of the ideally conducting polyhedron, the integral vector equation relative to the three constituents of the surface current reduces to a system of two scalar integral equations relative to the corresponding components of the current, expressed in the local coordinate systems.     

Mean square stability of stochastic Volterra integro-differential equations

The mean square stability of a non-linear stochastic Volterra integro-differential equation is studied. Non-convolution Volterra terms arise in both the drift and the dispersion term. Moreover, for the convolution case we determine the rate of convergence in terms of an integrability condition on the Volterra kernels.     

Matrix algorithms for solving (in)homogeneous bound state equations

In the functional approach to quantum chromodynamics, the properties of hadronic bound states are accessible via covariant integral equations, e.g. the Bethe–Salpeter equation for mesons. In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input. Analogously, inhomogeneous equations can be constructed to obtain off-shell information in addition to bound-state masses and other properties obtained from the covariant analogue to a wave function of the bound state. These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case). We demonstrate this by solving the homogeneous and inhomogeneous Bethe–Salpeter equations and find, e.g. that for the calculation of the mass spectrum it is as efficient or even advantageous to use the inhomogeneous equation as compared to the homogeneous. This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems.     

线性多智能体系统在固定和切换拓扑下的领航跟随控制

研究线性多智能体系统的领航跟随一致性问题. 假设每个多智能体系统只能得到其邻域的输出测量信息, 在此条件下, 讨论多智能体在有向固定网络拓扑和无向切换网络拓扑两种情况下的一致性问题. 针对这两种情况, 提出含有一种分布式观测器的一致性控制算法. 应用Lyapunov 稳定性理论证明了若单个多智能体系统是可镇定和可检测的, 且网络连接拓扑只需满足简单的结构, 则系统能够达到领航跟随一致性. 仿真结果验证了理论分析的正确性.     

New unconditionally stable difference schemes for the solution of multi-dimensional telegraphic equations

New unconditionally stable implicit difference schemes for the numerical solution of multi-dimensional telegraphic equations subject to appropriate initial and Dirichlet boundary conditions are discussed. Alternating direction implicit methods are used to solve two and three space dimensional problems. The resulting system of algebraic equations is solved using a tri-diagonal solver. Numerical results are presented to demonstrate the utility of the proposed methods.     

On a fallacious conjecture about the stabilizability properties of solutions of the Riccati difference equation

In [1], among other results, some conjectures concerning the monotonicity and stabilizing properties of solutions of the difference Riccati equation were proven to be fallacious by means of suitable counterexamples. Moreover, a ‘possibly fallacious conjecture’ was formulated for which no counterexample had been found. In this letter, such a counterexample is provided together with an interpretation of the somewhat counterintuitive behaviour of the Riccati equation.     

On the quantum master equation under feedback control

The nature of the quantum trajectories, described by stochastic master equations, may be jump-like or diffusive, depending upon different measurement processes. There are many different unravelings corresponding to different types of stochastic master equations for a given master equation. In this paper, we study the relationship between the quantum stochastic master equations and the quantum master equations in the Markovian case under feedback control. We show that the corresponding unraveling no longer exists when we further consider feedback control besides measurement. It is due to the fact that the information gained by the measurement plays an important role in the control process. The master equation governing the evolution of ensemble average cannot be restored simply by eliminating the noise term unlike the case without a control term. By establishing a fundamental limit on performance of the master equation with feedback control, we demonstrate the differences between the stochastic master equation and the master equation via theoretical proof and simulation, and show the superiority of the stochastic master equation for feedback control.