An approximate analytic solution of the nonlinear Riccati differential equation

In this paper, a hybrid method which combines the Adomian decomposition method (ADM), the Laplace transform algorithm and the Padé approximant is introduced to solve the approximate analytic solutions of the nonlinear Riccati differential equations. This hybrid method demonstrates accurate and reliable results, and has a great improvement in the ADM truncated series solution which diverges rapidly as the applicable domain increases. Three examples herein are given to demonstrate good accuracy and fast convergence in comparison with the exact solution.     

Dual-hop relaying communications over generalized-K (KG) fading channels

In this paper, a study on the end-to-end performance of dual-hop non-regenerative relaying over independent generalized-K (KG) fading channels is presented. Using a suitable upper bound for the end-to-end signal-to-noise ratio (SNR), novel closed-form expressions for the cumulative distribution function (CDF), probability density function (PDF) and the moments of this bounded SNR are derived. These results can be afterwards used to obtain important performance metrics of the considered system such as the outage probability and the error performance of digital modulation schemes. In the case of independent but non-necessarily identical fading channels, lower bounds for the average bit error probability (ABEP) for different modulation schemes are determined by using the Padé approximants method. For the case of identical fading channels, closed-form lower bounds for the ABEP are derived. Various numerical and computer simulation results illustrate the proposed analysis.     

Convection-radiation from a continuously moving fin of variable thermal conductivity

The homotopy analysis method (HAM) is applied to generate an analytic solution for heat transfer in a moving fin of variable thermal conductivity which is losing heat by simultaneous convection and radiation to its surroundings. The accuracy of the analytic solution is validated by comparing it with the direct numerical solution of the problem. The analytic solution is found to be accurate to at least three places of decimal for a wide range of values of the parameters that are commonly encountered in thermal processing application. Graphs displaying the results are interpreted in physical terms.     

The graph-theoretic field model—I. Modelling and formulations

The Graph-Theoretical Field Model provides a unifying approach for developing numerical models of field and continuum problems. The methodology examines the field problem from the first stages of conceptualization without recourse to the governing differential equations of the field problem; this is accomplished by deriving discrete statements of the physical laws which govern the field behaviour. There are generally three laws, and these are modelled by the “cutset equations”, the “circuit equations”, and the “terminal equations”. In order to establish these three sets of equations it is expedient first to spatially discretize the field in a manner similar to the finite difference method and then to associate a linear graph (denoted as the field graph) with the spatial discretization. The concept of “through” and “across” variables, which underlies the cutset and circuit equations respectively, enables one to define the graph in an unambiguous manner such that each “edge” of the graph identifies a pair of complementary variables. From a knowledge of the constitutive properties and the boundary conditions of the field it is possible to associate terminal equations with sets of edges. Since the resulting sets of equations represent the field equations, these equations provide the basis for a complete (but approximate) solution to the field or continuum problem. In fact, this system approach uses a two part model: one for the components and another for the interconnection pattern of the components which renders the formulation procedures totally independent of the solution procedure.This paper presents the theoretical basis of the model and several graph-theoretic formulations for steady-state problems. Examples from heat conduction and small- deformation elasticity are included.     

On optimal location and management of a new industrial plant: Numerical simulation and control

Within the framework of numerical modelling and multi-objective control of partial differential equations, in this work we deal with the problem of determining the optimal location of a new industrial plant. We take into account both economic and ecological objectives, and we look not only for the optimal location of the plant but also for the optimal management of its emissions rate. In order to do this, we introduce a mathematical model (a system of nonlinear parabolic partial differential equations) for the numerical simulation of air pollution. Based on this model, we formulate the problem in the field of multi-objective optimal control from a cooperative viewpoint, recalling the standard concept of Pareto-optimal solution, and pointing out the usefulness of Pareto-optimal frontier in the decision making process. Finally, a numerical algorithm – based on a characteristics/Galerkin discretization of the adjoint model – is proposed, and some numerical results for a hypothetical situation in the region of Galicia (NW Spain) are presented.     

二阶系统数值解耦方法的研究

数值代数领域通过保持Lancaster结构来研究二阶系统的解耦问题,但寻找解耦变换涉及到了非线性方程组求解问题,难以实现. 提出了一种二阶系统数值解耦的新方法. 根据系统解耦前后的同谱信息确定解耦后的系统,将寻找解耦变换的非线性问题转化为齐次Sylvester方程求解问题; 并利用矩阵的Kronecker积理论求解二阶系统的解耦变换. 数值试验证明了该方法的可行性,为二阶系统的数值解耦找到了更便易的实现途径.     

Least squares Hermitian solution of the complex matrix equation AXB+CXD=E with the least norm

In this paper, a new method is presented to solve the least squares Hermitian problem of the complex matrix equation AXB+CXD=E. The explicit expression of least squares Hermitian solution with the least norm is given. The least squares Hermitian solution with the least norm of complex matrix equation AXB=E is also derived. Numerical algorithms and numerical examples show the feasibility of our method.     

On stresses in a perforated strip

The convergence of the successive approximations in Howland's solution of an infinite perforated strip containing a symmetrically-located circular hole under longitudinal tension can be improved considerably by using the well known solution of an infinite perforated plate as the approximation of zero order. Further, by basing on a somewhat different physical concept, a corresponding direct solution of the problem can be formulated. Such a solution is presented in this paper. Numerical results are obtained from the solution over a wider range of values of the radius of hole, a feature hardly attainable in the previously known solutions of the problem. Further extension of the method of solution to the transverse bending problem as well as to the case of an unsymmetrically-located hole is also possible.     

Solution of the two-dimensional heat equation for a square in terms of elliptic functions

In this study, an alternative method is presented for the solution of two-dimensional heat equation in a square region. In this method, the solution function of the problem is based on the Green function, and therefore on elliptic functions. The numerical results have been computed for an application example and compared with the results of the well-known method of separation of variables and the finite elements method. The comparison study shows that the present method gives better results than those of the other methods.     

中等职业学校体育现状研究与分析

由于中等职业学校体育教育观念的滞后,导致了学校体育工作的落后。普遍存在体育教师数量不足、质量下降和工作态度不端正,学生体育情感淡薄等一系列问题,严重影响了学校体育的发展。本文通过对这些问题进行分析与研究,探讨解决这些问题的途径。     

Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials

In this paper, a numerical method to solve nonlinear optimal control problems with terminal state constraints, control inequality constraints and simple bounds on the state variables, is presented. The method converts the optimal control problem into a sequence of quadratic programming problems. To this end, the quasilinearization method is used to replace the nonlinear optimal control problem with a sequence of constrained linear-quadratic optimal control problems, then each of the state variables is approximated by a finite length Chebyshev series with unknown parameters. The method gives the information of the quadratic programming problem explicitly (The Hessian, the gradient of the cost function and the Jacobian of the constraints). To show the effectiveness of the proposed method, the simulation results of two constrained nonlinear optimal control problems are presented.     

A composite pseudospectral method for solving multi-delay optimal control problems involving piecewise constant delay functions

An adaptive numerical method for solving multi-delay optimal control problems with piecewise constant delay functions is introduced. The proposed method is based on composite pseudospectral method using the well-known Legendre–Gauss–Lobatto points. In this approach, the main problem converts to a mathematical optimization problem whose solution is much more easier than the original one. The necessary conditions of optimality associated to nonlinear piecewise constant delay systems are derived. The method is easy to implement and provides very accurate results.     

基于熵值法的我国省际生态数据评价研究

【目的/意义】构建科学的生态数据评价模型是生态文明建设的重要内容,需要从针对性、严谨性入手对数 据进行归纳分析,为提高评价体系的准确性提供有力数据支撑。【方法/过程】本文以我国省际生态指数为例,运用 熵值法对评价指标、权重进行优化,通过数据模型的构建,从海量数据中筛选出能够科学、精准地评价我国省际当 前生态文明建设现状的数据指标。【结果/结论】基于熵值法的评价模型,合理地解决了评价体系评价不精准、不符合 实际的问题,展示了省际生态发展情况,提出了我国省际生态发展评价问题的解决途径,实证研究的结果表明该评 价方法是科学的。【创新/局限】本文利用改进的熵值法,得到我国省际生态指数数据评价模型,用于后续的综合评 价。鉴于数据的可获得性,部分研究采取了数值模拟方法,未来要完善计量模型分析,进一步提高结果的精准度。     

半导体器件模拟方法研究

主要介绍了半导体器件模拟中的几种物理模型和三角形网格自适应划分的算法。并以泊松方程为例,介绍了如何在网格中进行方程的离散化,以及后期计算方程组的几种数值方法并比较各种算法的优缺点,同时得到解的示意图。     

Symmetric least squares solution of a class of Sylvester matrix equations via MINIRES algorithm

The purpose of this paper is deriving the minimal residual (MINIRES) algorithm for finding the symmetric least squares solution on a class of Sylvester matrix equations. We prove that if the system is inconsistent, the symmetric least squares solution can be obtained within finite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrix to obtain the minimum norm least squares symmetric solution of the problem. Finally, we give some numerical examples to illustrate the performance of MINIRES algorithm.     

An adaptive mesh method with variable relaxation time

Moving mesh partial differential equations have been widely used in the last decade for solving differential equations exhibiting large solution variations such as shock waves and boundary layers.In this paper, we have applied a dynamic adaptive method for solving time-dependent differential equations. The mesh velocities are governed by an equation in which a relaxation time is employed to move nodes in such a way that they remain concentrated in regions of rapid variation of the solution. A numerical example involving a blow-up problem shows the advantage of using a variable relaxation time over a fixed one.     

计算流体力学在搅拌混合过程模拟中的应用

计算流体力学包括反应器中流体的质量、动量、能量守恒方程和反映所研究问题的辅助方程的数值解。它在搅拌式反应器的设计与放大中起着重要的作用。概述了模拟搅拌式反应器中流场的各种方法:搅拌桨边界条件法,内外迭代法,多重参照系法,滑动网格法,快速照相法和大涡模拟法。并通过分析各种方法的优缺点提出了进一步的研究方向:修正湍流模型,发展大涡模拟法等。     

Adaptive pseudospectral methods for solving constrained linear and nonlinear time-delay optimal control problems

In this paper, we first develop an adaptive shifted Legendre–Gauss (ShLG) pseudospectral method for solving constrained linear time-delay optimal control problems. The delays in the problems are on the state and/or on the control input. By dividing the domain of the problem into a uniform mesh based on the delay terms, the constrained linear time-delay optimal control problem is reduced to a quadratic programming problem. Next, we extend the application of the adaptive ShLG pseudospectral method to nonlinear problems through quasilinearization. Using this scheme, the constrained nonlinear time-delay optimal control problem is replaced with a sequence of constrained linear-quadratic sub-problems whose solutions converge to the solution of the original nonlinear problem. The method is called the iterative-adaptive ShLG pseudospectral method. One of the most important advantages of the proposed method lies in the case with which nonsmooth optimal controls can be computed when inequality constraints and terminal constraints on the state vector are imposed. Moreover, a comparison is made with optimal solutions obtained analytically and/or other numerical methods in the literature to demonstrate the applicability and accuracy of the proposed methods.     

Computational burden reduction in min-max MPC

Min-max model predictive control (MMMPC) is one of the strategies used to control plants subject to bounded uncertainties. The implementation of MMMPC suffers a large computational burden due to the complex numerical optimization problem that has to be solved at every sampling time. This paper shows how to overcome this by transforming the original problem into a reduced min-max problem whose solution is much simpler. In this way, the range of processes to which MMMPC can be applied is considerably broadened. Proofs based on the properties of the cost function and simulation examples are given in the paper.     

An analytical solution to applied mathematics-related Love’s equation using the Boubaker polynomials expansion scheme

In this paper, we present a method for the numerical solution of Love's equation in a particular physical system. The resolution protocol is based on the Boubaker Polynomials Expansion Scheme (BPES), followed by array analyses. The obtained results are compared with some recently published ones. The accuracy and the asymptotic behaviors of the solutions are discussed.