Optimal control of discrete-time singularly perturbed systems

In this paper, we present a new algorithm for solving the optimal control of discrete-time singularly perturbed systems. The main idea of this algorithm is based on two steps. First, the Hamiltonian difference equation is reduced to the backward recursive form rather than the forward recursive form. Second, the bilinear transformation is applied to transform the derived non-symmetric discrete-time Riccati equations into continuous-time equations. In order to improve the efficiency of this scheme, two matrix permutations are introduced into this algorithm by taking into account the previous work of Gajic and Shen (1991). Therefore, substantial numerical advantages are gained; namely, computation and memory requirements. The F-8 aircraft model is used to illustrate the efficiency of the proposed method.     

The Fourth-order Difference Equation Satisfied by the Associated Orthogonal Polynomials of the Δ -Laguerre–Hahn Class

Starting from the D_ω-Riccati difference equation satisfied by the Stieltjes function of a linear functional, we work out an algorithm which enables us to write the general fourth-order difference equation satisfied by the associated of any integer order of orthogonal polynomials of the Δ -Laguerre–Hahn class. Moreover, in classical situations (Meixner, Charlier, Krawtchouk and Hahn), we give these difference equations explicitly; and from the Hahn difference equation, by limit processes we recover the difference equations satisfied by the associated of the classical discrete orthogonal polynomials and the differential equations satisfied by the associated of the classical continuous orthogonal polynomials.     

Liouvillian solutions of third order differential equations

The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show that, except for four finite groups and a reduction to the second order case, it is possible to give a formula in the imprimitive case. We also give necessary conditions and several simplifications for the computation of the minimal polynomial for the remaining finite set of finite groups (or any known finite group) by extracting ramification information from the character table. Several examples have been constructed, illustrating the possibilities and limitations.     

An algorithm for the solution of second order fuzzy initial value problems

In this paper, we state a fuzzy initial value problem of the second order fuzzy differential equations. Here we investigate problems with fuzzy coefficients, fuzzy initial values and fuzzy forcing functions. We propose an algorithm based on alpha-cut of a fuzzy set. Finally we present some examples by using our proposed algorithm.     

High-order compact boundary value method for the solution of unsteady convection–diffusion problems

In this paper, we propose a new class of high-order accurate methods for solving the two-dimensional unsteady convection–diffusion equation. These techniques are based on the method of lines approach. We apply a compact finite difference approximation of fourth order for discretizing spatial derivatives and a boundary value method of fourth order for the time integration of the resulted linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Also this method is unconditionally stable due to the favorable stability property of boundary value methods. Numerical results obtained from solving several problems include problems encounter in many transport phenomena, problems with Gaussian pulse initial condition and problems with sharp discontinuity near the boundary, show that the compact finite difference approximation of fourth order and a boundary value method of fourth order give an efficient algorithm for solving such problems.     

Solving fuzzy equations using evolutionary algorithms and neural nets

 In this paper we use evolutionary algorithms and neural nets to solve fuzzy equations. In Part I we: (1) first introduce our three solution methods for solving the fuzzy linear equation AˉXˉ + Bˉ= Cˉ; for Xˉ and (2) then survey the results for the fuzzy quadratic equations, fuzzy differential equations, fuzzy difference equations, fuzzy partial differential equations, systems of fuzzy linear equations, and fuzzy integral equations; and (3) apply an evolutionary algorithm to construct one of the solution types for the fuzzy eigenvalue problem. In Part II we: (1) first discuss how to design and train a neural net to solve AˉXˉ + Bˉ= Cˉ for Xˉ and (2) then survey the results for systems of fuzzy linear equations and the fuzzy quadratic.     

An extension of the Prelle-Singer method and a Maple implementation

The Prelle-Singer method is a semi-decision algorithm which can be used to solve analytically first order ordinary differential equations which have solutions in terms of elementary functions. In this paper we develop an extension to the Prelle-Singer method which deals with first order ordinary differential equations whose solutions lie outside the scope of the standard Prelle-Singer method. We present a software package in Maple V, Release 5 which implements both the Prelle-Singer method in its original form and our extension. Tests with ordinary differential equations taken from standard references show that our package is able to solve equations where Maple's standard solution routines fail.     

求解Maxwell线性棱元鞍点系统的并行Uzawa算法

本文针对一类Maxwell方程组鞍点问题的第一类N啨d啨lec线性棱元离散系统,设计了一种基于节点辅助空间预条件子的并行Uzawa算法(HX-Uzawa-p)。数值实验结果表明,不论是对光滑系数还是对有无浮动子区域及有无内交叉点的跳系数情形,我们所设计的并行算法HX-Uzawa-p的迭代次数都基本不依赖于网格规模及系数跳幅,且具有很好的并行可扩展性。     

A highly accurate alternating 6-point group method for the dispersive equation

In this paper, we construct a group of Saul'yev type asymmetric difference formulas for the dispersive equation. Based on these formulas we derive a new alternating 6-point group algorithm to solve dispersive equations with periodic boundary conditions. The algorithm has a high-order accuracy in space and an unconditional stability. The theoretical results are conformed to the numerical simulation. A comparison of this algorithm with the previous Alternating Group Explicit method is presented.     

On the numerical solution of two coupled nonlinear partial integro-differential equations related to the optimal control of a nonlinear noisy oscillator

This paper deals with the optimal control of a nonlinear stochastic thrid-order oscillator. It is shown that, in order to implement the optimal feedback control law, a set of two coupled nonlinear partial integro-differential equations has to be solved. A finite difference algorithm for the solution of these equations is proposed, and its efficiency and applicability are demonstrated with examples.     

Systolic arrays for group explicit methods for solving first order hyperbolic equations

In this paper, systolic techniques for the simulation of the Group Explicitmethod for parabolic equations by systolic array data structures using the Soft Systolic algorithm paradigm have been extended to include hyperbolic equations of first order. In particular, the Group Explicit Complete (GEC) strategy can be used as generic array construction for other simple schemes and for comput ations on large intervals. The easy form of the finite difference approximation to the hyperbolic equation res ults in a simpler and hence faster and more area efficient basic cell.     

A high-order compact finite difference method and its extrapolation for fractional mobile/immobile convection–diffusion equations

This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection–diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm.     

A method of inexact steepest descent for systems of linear equations

Our approach combines the method of inexact steepest descent with the method of contractor directions to obtain an algorithm for solving systems of linear equations. In order to enhance the scope of applicability, we consider an iterative method with variable step-size iterations. We prove the convergence and given an error estimate for our method.

The algorithm is well-suited for parallel computation. In fact, for systems with m equations and n unknowns, each iteration may be computed in parallel time O(log m + log n), on an EREW PRAM with O(mn) processors.     

Simulations of Shallow Water Equations with Finite Difference Lax-Wendroff Weighted Essentially Non-oscillatory Schemes

In this paper we study a Lax-Wendroff-type time discretization procedure for the finite difference weighted essentially non-oscillatory (WENO) schemes to solve one-dimensional and two-dimensional shallow water equations with source terms. In order to maintain genuinely high order accuracy and suit to problems with a rapidly varying bottom topography we use WENO reconstruction not only to the flux but also to the source terms of algebraical modified shallow water equations. Extensive simulations are performed, as a result, the WENO schemes with Lax-Wendroff-type time discretization can maintain nonoscillatory properties and more cost effective than that with Runge-Kutta time discretization.     

A new fractional Chebyshev FDM: an application for solving the fractional differential equations generated by optimisation problem

In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method. The algorithm is based on a combination of the useful properties of Chebyshev polynomial approximation and finite difference method. We implement this technique to solve numerically the non-linear programming problem which are governed by fractional differential equations (FDEs). The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the Caputo fractional derivatives. This operational matrix method can be regarded as a non-uniform finite difference scheme. The error bound for the fractional derivatives is introduced. The application of the method to the generated FDEs leads to algebraic systems which can be solved by an appropriate method. Two numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method. A comparison with the fourth-order Runge–Kutta method is given.     

解非线性方程组的多目标优化进化算法

如何有效地求解复杂非线性方程组是进化计算领域一个新的研究问题。将非线性方程组等价地转化成多目标优化问题,同时设计了求解的多目标优化进化算法。为了提高算法的搜索能力及避免算法陷入局部最优,采用了自适应Levy变异进化算子和均匀杂交算子。计算机仿真表明该算法对非线性方程组的求解是有效的。     

Computing differential invariants of hybrid systems as fixedpoints

We introduce a fixedpoint algorithm for verifying safety properties of hybrid systems with differential equations whose right-hand sides are polynomials in the state variables. In order to verify nontrivial systems without solving their differential equations and without numerical errors, we use a continuous generalization of induction, for which our algorithm computes the required differential invariants. As a means for combining local differential invariants into global system invariants in a sound way, our fixedpoint algorithm works with a compositional verification logic for hybrid systems. With this compositional approach we exploit locality in system designs. To improve the verification power, we further introduce a saturation procedure that refines the system dynamics successively with differential invariants until safety becomes provable. By complementing our symbolic verification algorithm with a robust version of numerical falsification, we obtain a fast and sound verification procedure. We verify roundabout maneuvers in air traffic management and collision avoidance in train control and car control.     

解偏微分方程逆问题的并行遗传算法

遗传算法是解优化问题的有效算法。本文论述转换偏微分方程逆问题（对应的差分方程逆问题）成优化问题并且构造出解该优化问题的并行遗传算法。     

二元域大型稀疏矩阵向量乘的FPGA设计与实现

作为Wiedemannn算法的核心部分,稀疏矩阵向量乘是求解二元域上大型稀疏线性方程组的主要步骤。提出了一种基于FPGA的二元域大型稀疏矩阵向量乘的环网硬件系统架构,为解决Wiedemannn算法重复计算稀疏矩阵向量乘,提出了新的并行计算结构。实验分析表明,提出的架构提高了Wiedemannn算法中稀疏矩阵向量乘的并行性,同时充分利用了FPGA的片内存储器和吉比特收发器,与目前性能最好的部分可重构计算PR模型相比,实现了2.65倍的加速性能。     

并行PCG算法在电法勘探中的应用研究

采用有限元法进行电法勘探时，会产生大型稀疏线性方程组，如何提高方程组的求解效率成为物探研究的关键。针对传统直接法难以实现并行求解的缺点，提出了在Beowulf集群环境下，采用并行PCG算法求解物探系统线性方程组。在集群环境下，该算法具有机器间相互通讯少、时间复杂度低等优点，并且易于并行实现。实验结果表明，采用PCG算法获得了良好的并行效果。