A miscellany of results of an equation of Count J. F. Riccati

A collection of results on the Riccati equation is presented. The questions addressed are the existence of strong solutions of the algebraic Riccati equation and the convergence of solutions of the Riccati difference equation to those of the algebraic equation. The results derived utilize detestability conditions only.     

A Neural Network Study of Blasius Equation

In this work we applied a feed forward neural network to solve Blasius equation which is a third-order nonlinear differential equation. Blasius equation is a kind of boundary layer flow. We solved Blasius equation without reducing it into a system of first order equation. Numerical results are presented and a comparison according to some studies is made in the form of their results. Obtained results are found to be in good agreement with the given studies.

     

具有变动边界的过渡面的热传导方程构造法

提出过渡面的热传导方程构造方法。使用热传导方程、初值条件和变动的边界条件,得到过渡面的偏微分方程模型,再利用分离变量法或者数值方法得到模型的解,则方程的像空间就是满足连续性要求的过渡面。数值实验也表明这种方法是可行的、有效的。     

旋转圆盘的振动主动控制

对工业中应用广泛的悬臂回转圆盘, 推导了悬 臂回转圆盘的横向运动方程,并将之演化为实模态状态空间的动态方程. 引入智能积分控制 ,改良了现有控制方法．在实模态状态空间,考虑对旋转圆盘横向振幅构成主要影响的几阶 模态,分别用两种方法进行仿真,仿真结果说明新方法可以得到更好的控制效果.     

Proper Generalized Decomposition Method for Solving Fisher-Type Equation and Heat Equation

A model reduction technique—the Proper Generalized Decomposition (PGD) for solving time dependent and multidimensional parameters is reviewed and applied to both the Fisher-type equations and the heat equation. Space-time discretization and separated representation technique for obtaining fast convergence computation while maintaining real time is detailed. Three situations of the Fisher-type equation are solved by the PGD and the results show a perfect agreement with the exact solutions. The source term of the heat equation is given a Huxley source and the thermal diffusivity is taken to be linearly dependent of the spatial parameter. The results show how the Fisher-type equation finds application to the heat equation and that the PGD method allows a perfect representation of the temperature distribution defined in a 5-D tensorial product space and time.     

Numerical convergence for the Bellman equation of stochastic optimal control with quadratic costs and constraints

A predictor-corrector, Crank-Nicolson computer algorithm is examined for the Bellman equation of stochastic optimal control with quadratic costs and constrained control. A linearized comparison equation is heuristically derived for the nonlinear and discontinuous Bellman equation. Convergence of the method is studied using von Neumann's Fourier stability method. A mesh-ratio-type condition for the convergence is derived for the comparison equation. This condition is uniform for both parabolic and hyperbolic versions of the nonlinear equation. The results are valid for Gaussian stochastic noise and Poisson noise.     

Riccati differential equation in optimal filtering of periodic non-stabilizable systems

This paper discusses the periodic solutions of the matrix Riccati differential equation in the optimal filtering of periodic systems. Special emphasis is given to non-stabilizable systems and the question addressed is the existence and uniqueness of a steady-state periodic non-negative definite solution of the periodic Riccati differential equation which gives rise to an asymptotically stable steady-state filter. The results presented show that the stabilizability is not a necessary condition for the existence of such a periodic solution. The convergence of the general solution of the periodic Riccati differential equation to a periodic equilibrium solution is also investigated. The results are extensions of existing time-invariant systems results to the case of periodic systems     

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.     

Finite difference solutions of dispersive partial differential equations

The second order leapfrog method is used to discretize the linearized KdV equation which is itself a dispersive partial differential equation. The resulting difference equation is solved and analyzed in terms of its dispersion relation and propagation properties. Numerical experiments are included to illustrate some of the results obtained. Finally, a novel aliasing error which affects the propagation of wave packets is presented     

Painlevé方程解的渐近性态的数值分析方法

Painlevé方程是六类重要的二阶代数微分方程,它们的发展一直受到人们的关注.解的渐近性态是重要的研究方向.由于解的渐近性态难以直接观察出来,所以我们用微分方程数值解研制出Painlevé方程解的渐近性态的分析系统.通过此系统对Painlevé方程解的渐近性态进行分析,已经得到一些结果,部分结果与有关文献的结果[1]相当吻合,进而为从理论上找出具体的相关性质提供了方法和依据.     

一类偏微分方程的格子Boltzmann模型

研究了对流扩散方程、Burgers方程和Modified-Burgers方程等具有相同形式的一类偏微分方程。并且构建了带修正函数项的D1Q3格子Boltzmann模型求解这类方程。为了能准确地恢复出此宏观方程,利用Chapman-Enskog展开和多尺度分析技术,推导出了各个方向的平衡态分布函数和修正函数的具体表达式。数值计算结果表明该模型是稳定、有效的。     

Correlated noise filtering and invariant directions for the Riccati equation

The Riccati equation associated with a class of discrete-time correlated noise problems is examined, and the concept of invariant directions for this equation is introduced. For single-output systems the set of such directions is completely characterized. Deletion of these directions by an appropriate transformation of the Riccati equation results in a minimal order equation for computation. This transformation also reveals the underlying structure of the optimal filter for the correlated noise problem.     

Nonlinear vibrations of cross-ply plates

Large amplitude free vibrations of simply supported rectangular cross-ply plates are examined. The restoring force function in the equation of motion is a combination of quadratic and cubic terms. The equation of motion is integrated using a Runge-Kutta numerical scheme of the MATLAB software. The results obtained are in good agreement with the existing results.     

Error function scheme for Burgers' equation

The present paper is devoted to the development of a new scheme to solve the one-dimensional time-dependent Burgers' equation locally on sub-domains, using similarity reductions for partial differential equations. Each sub-domain is divided into three grid points. The ordinary differential equation deduced from the similarity reduction can be integrated and is then used to approximate the flux vector in the Burgers' equation. The arbitrary constants in the analytical solution of the similarity equation can be determined in terms of the dependent variables at the grid points in each sub-domain. This approach eliminates the difficulties associated with boundary conditions for the similarity reductions over the whole solution domain. Numerical results are obtained for two different test cases and are compared with other numerical results.     

Hybrid Hamilton–Jacobi–Poisson wall distance function model

Expensive to compute wall distances are used in key turbulence models and also for the modeling of peripheral physics. A potentially economical, robust, readily parallel processed, accuracy improving, differential equation based distance algorithm is described. It is hybrid, partly utilising an approximate Poisson equation. This also allows auxiliary front propagation direction/velocity information to be estimated, effectively giving wall normals. The Poisson normal can be used fully, in an approximate solution of the eikonal equation (the exact differential equation for wall distance). Alternatively, a weighted fraction of this Poisson front direction (effectively, front velocity, in terms of the eikonal equation input) information and that implied by the eikonal equation can be used. Either results in a hybrid Poisson–eikonal wall distance algorithm. To improve compatibility of wall distance functions with turbulence physics a Laplacian is added to the eikonal equation. This gives what is termed a Hamilton–Jacobi equation. This hybrid Poisson–Hamilton–Jacobi approach is found to be robust on poor quality grids. The robustness largely results from the elliptic background presence of the Poisson equation. This elliptic component prevents fronts propagated from solid surfaces, by the hyperbolic eikonal equation element, reflecting off zones of rapidly changing grid density. Where this reflection (due to poor grid quality) is extreme, the transition of front velocity information from the Poisson to Hamilton–Jacobi equation can be done more gradually. Consistent with turbulence modeling physics, under user control, the hybrid equation can overestimate the distance function strongly around convex surfaces and underestimate it around concave. If the former trait is not desired the current approach is amenable to zonalisation. With this, the Poisson element is automatically removed around convex geometry zones.     

Analytical solution of gas lubricated slider microbearing

The analytical solution of a two-dimensional, isothermal, compressible gas flow in a slider microbearing is presented. A higher order accuracy of the solution is achieved by applying the boundary condition of Kn ² order for the velocity slip on the wall, together with the momentum equation of the same order (known as the Burnett equation). The analytical solution is obtained by the perturbation analysis. The order of all terms in continuum and momentum equations and in boundary conditions is evaluated by incorporating the exact relation between the Mach, Reynolds and Knudsen numbers in the modelling procedure. Low Mach number flows in microbearing with slowly varying cross-sections are considered, and it is shown that under these conditions the Burnett equation has the same form as the Navier–Stokes equation. Obtained analytical results for pressure distribution, load capacity and velocity field are compared with numerical solutions of the Boltzmann equation and some semi-analytical results, and excellent agreement is achieved. The model presented in this paper is a useful tool for the prediction of flow conditions in the microbearings. Also, its results are the benchmark test for the verifications of various numerical procedures.     

State-space solution of the causal factorization problem

Given rational matrix functions, G, H, and R, a necessary and sufficient condition is given for the existence of a causal solution to the equation GXH=R. A formula is given for a solution of the equation when it exists, and this particular solution is used to parameterize the set of all causal solutions of the equation. The results are specialized to the case when the function G is identically equal to identity. Applications of the author's results include exact model matching and disturbance decoupling problems     

广义Burgers方程的对称分类及其约化

利用李群方法对广义Burgers方程ut+f(x,t)(ux-uxx)=0的对称分类及其约化作具体讨论,其中f是关于自变量x,u的光滑函数,得到了f(x,t)的八种分类对称及相应的约化方程.该结果对于广义Burgers方程精确解的研究有重要意义.     

Design and analysis of microstrip‐line‐coupled NRD guide filter based on integral equation techniques

Integral equation techniques which include an order‐reduced volume integral equation approach for NRD guide circuits and a generalized surface‐volume integral equation approach for planar‐NRD guide hybrid circuits have been presented in this article. A gap‐coupled NRD guide filter is designed based on the order‐reduced volume integral equation method. The two identical high‐quality microstrip‐line‐to‐NRD guide transitions are designed by means of the generalized surface‐volume integral equation approach. Finally, the optimized length of the NRD guide feed‐line is determined and a compact microstrip‐line‐coupled NRD filter is obtained. Simulated results are in good agreement with measured results. © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2009.     

Solution of Electromagnetic Scattering Problems Involving Three-Dimensional Homogeneous Dielectric Objects by the Single Integral Equation Method

The problem of electromagnetic scattering by a homogeneous dielectric object is usually formulated as a pair of coupled integral equations involving two unknown currents on the surface S of the object. In this paper, however, the problem is formulated as a single integral equation involving one unknown current on S. Unique solution at resonance is obtained by using a combined field integral equation. The single integral equation is solved by the method of moments using a Galerkin test procedure. Numerical results for a dielectric sphere are in good agreement with the exact results. Furthermore, the single integral equation method is shown to have superior convergence speed of iterative solution compared with the coupled integral equations method.