Legendre-Weighted Residual Methods for System of Fractional Order Differential Equations

The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.

A FIELD METHOD FOR SOLVING THE EQUATIONS OF MOTION OF NONHOLONOMIC SYSTEMS

In this paper,the field method for solving the equations of motion of holonomic nonconservative systems is extended to nonholonomic systems with constant mass and with variable mass.Two examples are given to illustrate its application.

The method of variation on parameters for integration of a generalized Birkhoffian system

This paper focuses on studying the integration method of a generalized Birkhoffian system.The method of variation on parameters for the dynamical equations of a generalized Birkhoffian system is presented.The procedure for solving the problem can be divided into two steps：the first step,a system of auxiliary equations is constructed and its general solution is given;the second step,the parameters are varied,and the solution of the problem is obtained by using the properties of generalized canonical transformation.The method of variation on parameters for the generalized Birkhoffian system is of universal significance,and we take a nonholonomic system and a nonconservative system as examples to illustrate the application of the results of this paper.

Modeling Method for Flexible Energy Behaviors in CNC Machining Systems

CNC machining systems are inevitably confronted with frequent changes in energy behaviors because they are widely used to perform various machining tasks. It is a challenge to understand and analyze the flexible energy behaviors in CNC machining systems. A method to model flexible energy behaviors in CNC machining systems based on hierarchical objected-oriented Petri net(HOONet) is proposed. The structure of the HOONet is constructed of a high-level model and detail models. The former is used to model operational states for CNC machining systems, and the latter is used to analyze the component models for operational states. The machining parameters having great impacts on energy behaviors in CNC machining systems are declared with the data dictionary in HOONet models. A case study based on a CNC lathe is presented to demonstrate the proposed modeling method. The results show that it is effective for modeling flexible energy behaviors and providing a fine-grained description to quantitatively analyze the energy consumption of CNC machining systems.

Preparation of hydro-sodalite from fly ash using a hydrothermal method with a submolten salt system and study of the phase transition process

Hydro-sodalites are zeolitic materials with a wide variety of applications.Fly ash is an abundant industrial solid waste,rich in silicon and aluminum,from which hydro-sodalite can be synthesized.However,traditional hydrothermal synthesis methods are complex and cannot produce high-purity products.Therefore,there is a demand for processing routes to obtain high-purity hydro-sodalites.In the present study,high-purity hydro-sodalite(90.2 wt%)was prepared from fly ash by applying a hydrothermal method to a submolten salt system.Samples were characterized by powder X-ray diffraction(XRD),scanning electron microscopy(SEM),thermogravimetry and differential thermal analysis(TG–DTA),and Fourier transform infrared spectroscopy(FTIR)to confirm and quantify conversion of the raw material into the product phase.Purity of the samples prepared with an H2O/Na OH mass ratio of 1.5 and an H2O/fly ash mass ratio of 10 was calculated and the conversion process of the product phase was studied.Crystallinity of the product was influenced more by the Na OH concentration,less by the H2O/fly ash mass ratio.The main reaction process of the system is that the Si O ions produced by dissolution of the vitreous body in the fly ash and Na+ions in the solution reacted on the destroyed mullite skeleton to produce hydro-sodalite.This processing route could help mitigate processing difficulties,while producing high-purity hydro-sodalite from fly ash.

Large Deflection Dynamic Response Analysis of Flexible Hull Beams by the Multibody System Method

In this paper the large deflection dynamic problems of Euler beams are investigated. The vibration control equations are derived based on the multibody system method. A numerical procedure for solving the resulting differential algebraic equations is presented on the basis of the Newmark direct integration method combined with the Newton-Raphson iterative method. The sub beams are treated as small deformation in the convected coordinate systems, which can greatly simplify the deformation description. The rigid motions of the sub beams are taken into account through the motions of the convected coordinate systems. Numerical ex- amples are carried out, where results show the effectiveness of the proposed method.

Poisson theory and integration method of Birkhoffian systems in the event space

This paper focuses on studying the Poisson theory and the integration method of a Birkhoffian system in the event space. The Birkhoff＇s equations in the event space are given. The Poisson theory of the Birkhoffian system in the event space is established. The definition of the Jacobi last multiplier of the system is given, and the relation between the Jacobi last multiplier and the first integrals of the system is discussed. The researches show that for a Birkhoffian system in the event space, whose configuration is determined by （2n ＋ 1） Birkhoff＇s variables, the solution of the system can be found by the Jacobi last multiplier if 2n first integrals are known. An example is given to illustrate the application of the results.

GENERALIZED VARIATIONAL DATA ASSIMILATION METHOD AND NUMERICAL EXPERIMENT FOR NON-DIFFERENTIAL SYSTEM

The generalized variational data assimilation for non-differential dynamical systems is studied.There is no tangent linear model for non-differential systems and thus the general adjoint model can not be derived in the traditional way.The weak form of the original system was introduced, and then the generalized adjoint model was derived. The generalized variational data assimilation methods were developed for non-differential low dimensional system and non-differential high dimensional system with global and local observations. Furthermore, ideas in inverse problems are introduced to 4DVAR (Four-dimensional variational) of non-differential partial differential system with local observations.

Design of Retarded Fractional Delay Differential Systems Using the Method of Inequalities

Methods based on numerical optimization are useful and effective in the design of control systems. This paper describes the design of retarded fractional delay differential systems （RFDDSs） by the method of inequalities, in which the design problem is formulated so that it is suitable for solution by numerical methods. Zakian＇s original formulation, which was first proposed in connection with rational systems, is extended to the case of RFDDSs. In making the use of this formulation possible for RFDDSs, the associated stability problems are resolved by using the stability test and the numerical algorithm for computing the abscissa of stability recently developed by the authors. During the design process, the time responses are obtained by a known method for the numerical inversion of Laplace transforms. Two numerical examples are given, where fractional controllers are designed for a time-delay and a heat-conduction plants.

Fluidized bed control system based on inverse system method

The invertible of the Large Air Dense Medium Fluidized Bed (ADMFB) were studied by introducing the concept of the inverse system theory of nonlinear systems. Then the ADMFB, which was a multivariable, nonlinear and coupled strongly system, was decoupled into independent SISO pseudo-linear subsystems. Linear controllers were designed for each of subsystems based on linear systems theory. The practice output proves that this method improves the stability of the ADMFB obviously.

The Relation between the Stabilization Problem for Discrete Event Systems Modeled with Timed Petri Nets via Lyapunov Methods and Max-Plus Algebra

A discrete event system is a dynamical system whose state evolves in time by the occurrence of events at possibly irregular time intervals. Timed Petri nets are a graphical and mathematical modeling tool applicable to discrete event systems in order to represent its states evolution where the timing at which the state changes is taken into consideration. One of the most important performance issues to be considered in a discrete event system is its stability. Lyapunov theory provides the required tools needed to aboard the stability and stabilization problems for discrete event systems modeled with timed Petri nets whose mathematical model is given in terms of difference equations. By proving stability one guarantees a bound on the discrete event systems state dynamics. When the system is unstable, a sufficient condition to stabilize the system is given. It is shown that it is possible to restrict the discrete event systems state space in such a way that boundedness is achieved. However, the restriction is not numerically precisely known. This inconvenience is overcome by considering a specific recurrence equation, in the max-plus algebra, which is assigned to the timed Petri net graphical model.

Noether＇s theorem and one-step corrections method for holonomic system

In this paper, a new computational method for improving the accuracy of numerically computed solutions is introduced. The computational method is based on the one-step method and conserved quantities of holonomic systems are considered as kinematical constraints in this method.

Novel method for random vibration analysis of single-degree-of-freedom vibroimpact systems with bilateral barriers

The vibroimpact systems with bilateral barriers are often encountered in practice.However,the dynamics of the vibroimpact system with bilateral barriers is full of challenges.Few closed-form solutions were obtained.In this paper,we propose a novel method for random vibration analysis of single-degree-of-freedom(SDOF)vibroim-pact systems with bilateral barriers under Gaussian white noise excitations.A periodic approximate transformation is employed to convert the equations of the motion to a con-tinuous form.The probabilistic description of the system is subsequently defined through the corresponding Fokker-Planck-Kolmogorov(FPK)equation.The closed-form station-ary probability density function(PDF)of the response is obtained by solving the reduced FPK equation and using the proposed iterative method of weighted residue together with the concepts of the circulatory probability flow and the potential probability flow.Finally,the versatility of the proposed approach is demonstrated by its application to two typical examples.Note that the solution obtained by using the proposed method can be used as the benchmark to examine the accuracy of approximate solutions obtained by other methods.

PARAMETRIC EQUATIONS OF NONHOLONOMIC NONCONSERVATIVE SYSTEMS IN THE EVENT SPACE AND THE METHOD OF THEIR INTEGRATION

In this paper,the parametric equations with multipliers of nonholonomic nonconservative sys- tems in the event space are established,their properties are studied,and their explicit formulation is obtained. And then the field method for integrating these equations is given.Finally,an example illustrating the appli- cation of the integration method is given.

APPLICATION OF THE MOMENT METHODS TO ANALYSIS OF X-RAY DIFFRACTION LINE PROFILE FOR PA1010-BMI SYSTEM

Pure X-ray diffraction profiles have been analysed for polyamide 1010 and PA1O1O-BMI system by means of multipeak fitting resolution of X-ray diffraction. The methods of variance and fourth moment have been applied to determine the particle size and strain values for the paracrystalline materials. The results indicated that both variance and fourth moment of X-ray diffraction line profile yielded approximately the same values of the particle size and the strain. The particle sizes of (100) reflection have been found to decrease with increasing BMI content, whereas the strain values increased.

SEISMIC RANDOM RESPONSE ANALYSIS OF HYSTERETIC SYSTEMS WITH PSEUDO EXCITATION METHOD

The stationary random responses of nonlinear shear-typeMulti-Degrees-of-Freedom (MDOF) hysteretic system are analyzed byusing the Pseudo Excitation Method (PEM) combined with the EquivalentLinerization Method (ELM). The solution of the equivalent linearsystem is obtained by iteratively solving complex algebraic equationsinstead of the Lyapunov equations. The efficiency of this method ismuch higher For practical engineering systems with manydegrees-of-freedom.

DNA extraction of birch leaves by improved CTAB method and optimization of its ISSR system

The basic method of DNA extraction （CTAB） was improved as the multi-times STE-CTAB extraction method and used to extract the DNA of birch leaved in this experiment. Results showed that the improved method is suitable not only for genomic DNA extraction of birch but also for that of other plants. The purity of genornic DNA extracted by the.multi-times STE-CTAB extraction method is higher than that by one time STE-CTAB method, and it does not need the process of RNase. The factors of influencing ISSR system were explored based on the genomic DNA of birch extracted by the two methods. The optimal conditions for ISSR system were determined as follows： Mg2＋ concentration is 1.5-3.0 mmol·L^-1, dNTP concentration 0.104）.25 mmol·L^-1, the quantity of Taq polymerase 0.5-2.0 U, template DNA 30-100 ng, and the concentration of primer is 0.2-0.4 pmmol·L^-1, and the reaction program was as： initial denaturation for 5 min at 94℃, 30 cycles of denaturation for 30 s at 94℃,annealing for 30 s at 51℃, extension for 30 s at 72℃, and a final 7 min extension at 72℃.

Sinc-Collocation Method for Solving Linear and Nonlinear System of Second-Order Boundary Value Problems

Sinc methods are now recognized as an efficient numerical method for problems whose solutions may have singularities, or infinite domains, or boundary layers. This work deals with the sinc-collocation method for solving linear and nonlinear system of second order differential equation. The method is then tested on linear and nonlinear examples and a comparison with B-spline method is made. It is shown that the sinc-collocation method yields better results.

A Hybrid Unit Commitment Approach Incorporating Modified Priority List with Charged System Search Methods

This paper presents a new hybrid approach that combines Modified Priority List (MPL) with Charged System Search (CSS), termed MPL-CSS, to solve one of the most crucial power system’s operational optimization problems, known as unit commitment (UC) scheduling. The UC scheduling problem is a mixed-integer nonlinear problem, highly-dimensional and extremely constrained. Existing meta-heuristic UC solution methods have the problems of stopping at a local optimum and slow convergence when applied to large-scale, heavily-constrained UC applications. In the first step of the proposed method, initial hourly optimum solutions of UC are obtained by Modified Priority List (MPL);however, the obtained UC solution may still be possible to be further improved. Therefore, in the second step, the CSS is utilized to achieve higher quality solutions. The UC is formulated as mixed integer linear programming to ensure the tractability of the results. The proposed method is successfully applied to a popular test system up to 100 units generators for both 24-hr and 168-hr system. Computational results show that both solution cost and execution time are superior to those of published methods.

Applications of Localized Phase Compensation Method to Design a Stabilizer in a Multi-machine Power System

为了演示和验证稳定器设计的就地相位补偿法在多机电力系统中的应用,介绍在多机电力系统中,就地补偿设计稳定器的2个应用实例。第1个实例是在多机电力系统中就地补偿设计电力系统稳定器（power system stabilizer,PSS）,阻尼电力系统局部模振荡。第2个实例是就地补偿设计附加在静态同步补偿器（static synchronous compensator,STATCOM）上的稳定器,抑制多机电力系统中的区域模振荡,并给出在一个16机电力系统中的应用计算和仿真结果。