Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

Comparative Study on Results of Euler,Improved Euler and Run­ge-Kutta Methods for Solving the Engineering Unknown Problems

The paper presents the comparative study on numerical methods of Euler method,Improved Euler method and fourth-order Runge-Kutta method for solving the engineering problems and applications.The three proposed methods are quite efficient and practically well suited for solving the unknown engineering problems.This paper aims to enhance the teaching and learning quality of teachers and students for various levels.At each point of the interval,the value of y is calculated and compared with its exact value at that point.The next interesting point is the observation of error from those methods.Error in the value of y is the difference between calculated and exact value.A mathematical equation which relates various functions with its derivatives is known as a differential equation.It is a popular field of mathematics because of its application to real-world problems.To calculate the exact values,the approximate values and the errors,the numerical tool such as MATLAB is appropriate for observing the results.This paper mainly concentrates on identifying the method which provides more accurate results.Then the analytical results and calculates their corresponding error were compared in details.The minimum error directly reflected to realize the best method from different numerical methods.According to the analyses from those three approaches,we observed that only the error is nominal for the fourth-order Runge-Kutta method.

Application of the Improved Kudryashov Method to Solve the Fractional Nonlinear Partial Differential Equations

Our purpose of this paper is to apply the improved Kudryashov method for solving various types of nonlinear fractional partial differential equations. As an application, the time-space fractional Korteweg-de Vries-Burger (KdV-Burger) equation is solved using this method and we get some new travelling wave solutions. To acquire our purpose a complex transformation has been also used to reduce nonlinear fractional partial differential equations to nonlinear ordinary differential equations of integer order, in the sense of the Jumarie’s modified Riemann-Liouville derivative. Afterwards, the improved Kudryashov method is implemented and we get our required reliable solutions where the results are justified by mathematical software Maple-13.

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

Stochastic Programming for Hub Energy Management Considering Uncertainty Using Two-Point Estimate Method and Optimization Algorithm

To maximize energy profit with the participation of electricity,natural gas,and district heating networks in the day-ahead market,stochastic scheduling of energy hubs taking into account the uncertainty of photovoltaic and wind resources,has been carried out.This has been done using a new meta-heuristic algorithm,improved artificial rabbits optimization(IARO).In this study,the uncertainty of solar and wind energy sources is modeled using Hang’s two-point estimating method(TPEM).The IARO algorithm is applied to calculate the best capacity of hub energy equipment,such as solar and wind renewable energy sources,combined heat and power(CHP)systems,steamboilers,energy storage,and electric cars in the day-aheadmarket.The standard ARO algorithmis developed to mimic the foraging behavior of rabbits,and in this work,the algorithm’s effectiveness in avoiding premature convergence is improved by using the dystudynamic inertia weight technique.The proposed IARO-based scheduling framework’s performance is evaluated against that of traditional ARO,particle swarm optimization(PSO),and salp swarm algorithm(SSA).The findings show that,in comparison to previous approaches,the suggested meta-heuristic scheduling framework based on the IARO has increased energy profit in day-ahead electricity,gas,and heating markets by satisfying the operational and energy hub limitations.Additionally,the results show that TPEM approach dependability consideration decreased hub energy’s profit by 8.995%as compared to deterministic planning.

The Zhou’s Method for Solving the Euler Equidimensional Equation

In this work, we apply the Zhou’s method [1] or differential transformation method (DTM) for solving the Euler equidimensional equation. The Zhou’s method may be considered as alternative and efficient for finding the approximate solutions of initial values problems. We prove superiority of this method by applying them on the some Euler type equation, in this case of order 2 and 3 [2]. The power series solution of the reduced equation transforms into an approximate implicit solution of the original equations. The results agreed with the exact solution obtained via transformation to a constant coefficient equation.

Improved-GRACE卫星重力轨道参数优化研究

基于改进的半解析法,利用激光干涉系统星间速度误差、GPS接收机轨道位置误差和轨道速度误差以及加速度计非保守力误差影响累计大地水准面的联合误差模型,开展了我国Improved-GRACE卫星重力测量计划轨道参数的优化选取论证。模拟结果表明:1)在300阶处,基于350km轨道高度估计累计大地水准面的精度为3.993×10-1m,基于300km和250km轨道高度估计精度分别提高了8.770倍和77.145倍,基于400km和450km轨道高度估计精度分别降低了8.718倍和75.307倍;2)基于50km星间距离估计累计大地水准面的精度为3.993×10-1m,基于110km和220km星间距离估计精度分别降低了1.259倍和1.395倍;3)我国将来首颗Improved-GRACE重力卫星的平均轨道高度和平均星间距离设计为350km与50km较优。

基于Euler/N-S方程的跨音速非线性静气动弹性问题研究

在C-H网格的基础上,采用Jam eson的中心差分有限体积法求解Eu ler/N-S方程,采用结构影响系数法计算结构的弹性变形,用三角元面积加权法和常体积转换法(CVT)实现流固耦合。

Euler-Bernoulli海洋立管涡致强迫振动响应研究

针对海洋立管(Pipe-in-pipe,PIP)系统在海水作用下发生的振动问题,开展了对PIP系统在涡致强迫振动下的动力学响应研究,分析了在涡致强迫振动下海洋立管外管直径、轴向拉力、外激力频率对海洋立管位移响应的影响规律。基于Euler-Bernoulli双梁模型,采用Lamb-Oseen涡模型,建立了动力学模型,利用格林函数法求得该强迫振动的稳态响应。结果表明,随着管道直径增加,外激力增加,产生最大力幅值的位置离管道越远;轴向拉力对外部管道的影响较大,对内部管道的影响较小;无因次频率取0.4时,外部管道位移超出允许变形极限,内外管壁发生周期碰撞,易对海洋立管造成损伤。

Consistency and Validity of the Mathematical Models and the Solution Methods for BVPs and IVPs Based on Energy Methods and Principle of Virtual Work for Homogeneous Isotropic and Non-Homogeneous Non-Isotropic Solid Continua

Energy methods and the principle of virtual work are commonly used for obtaining solutions of boundary value problems (BVPs) and initial value problems (IVPs) associated with homogeneous, isotropic and non-homogeneous, non-isotropic matter without using (or in the absence of) the mathematical models of the BVPs and the IVPs. These methods are also used for deriving mathematical models for BVPs and IVPs associated with isotropic, homogeneous as well as non-homogeneous, non-isotropic continuous matter. In energy methods when applied to IVPs, one constructs energy functional (I) consisting of kinetic energy, strain energy and the potential energy of loads. The first variation of this energy functional (δI) set to zero is a necessary condition for an extremum of I. In this approach one could use δI = 0 directly in constructing computational processes such as the finite element method or could derive Euler’s equations (differential or partial differential equations) from δI = 0, which is also satisfied by a solution obtained from δI = 0. The Euler’s equations obtained from δI = 0 indeed are the mathematical model associated with the energy functional I. In case of BVPs we follow the same approach except in this case, the energy functional I consists of strain energy and the potential energy of loads. In using the principle of virtual work for BVPs and the IVPs, we can also accomplish the same as described above using energy methods. In this paper we investigate consistency and validity of the mathematical models for isotropic, homogeneous and non-isotropic, non-homogeneous continuous matter for BVPs that are derived using energy functional consisting of strain energy and the potential energy of loads. Similar investigation is also presented for IVPs using energy functional consisting of kinetic energy, strain energy and the potential energy of loads. The computational approaches for BVPs and the IVPs designed using energy functional and principle of virtual work, their consistency and validity are also investigated. Classical continuum mechanics (CCM) principles i.e. conservation and balance laws of CCM with consistent constitutive theories and the elements of calculus of variations are employed in the investigations presented in this paper.

基于PSO-GRNN和D-S证据理论的电网分区故障诊断

针对大电网中保护和断路器误动、拒动、信息丢失等不确定的电网故障信息以及现有电网分区方法的不足,提出了基于粒子群优化广义回归神经网络(PSO-GRNN)和D-S证据理论的电网分区故障诊断方法。首先,通过改进图形分割法将大电网划分为相互重叠的不同区域,降低故障诊断难度。然后在各个区域建立PSOGRNN诊断模块,根据故障警报信息,并行完成各自的故障诊断任务。最后,采用D-S证据理论对相邻区域的重叠区域进行分析,以实现对重叠区域的综合故障诊断。仿真结果表明,该方法能有效识别非重叠区域和重叠区域的故障,容错能力强,诊断准确率高。

An Inexact Halley's Method

An inexact Halley＇s method-Halley-PCG（preconditioned conjugate gradient） method is proposed for solving the systems of linear equations for improved Halley method either by Cholesky factorization exactly or by preconditioned conjugate gradient method approximately. The convergence result is given and the efficiency of the method compared to the improved Halley＇s method is shown.

基于无人机点云与改进R_(S)表征法的结构面粗糙度定量分析

为研究采样尺寸、采样间距对高陡斜坡岩体结构面粗糙度评价结果的影响,选取藏东南某铁路察达车站工点左岸高陡岩质斜坡为研究区,提出采用无人机多角度贴近摄影测量技术获取毫米级点云数据以建立研究区斜坡岩体高精度三维模型,并从中选取典型区域裁剪出带有点云信息的27条面状结构面,使用Delaunay三角化原理对结构面进行网格化重建。基于此,提出一种采用点云拟合平面代替R_(S)表征法中垂直投影平面的新方法,并研究结构面粗糙度在不同采样尺寸、采样间距下的变化规律,结果表明:不同三角剖分方式对R_(S)表征值影响较小;结构面粗糙度具有尺寸效应与间距效应,其粗糙度表征值随结构面尺寸的增加逐渐趋于稳定,随结构面采样间距增大逐渐减小;部分存在尺寸效应的结构面存在“假有效采样尺寸”与“真有效采样尺寸”。在进行粗糙度评价时应确保所得有效采样尺寸为“真有效采样尺寸”。

A Numerical Study of Several Species Population Models

This work considers a special case of Lotka-Volterra equations, which means that in the system of two ordinary differential equations, we take the four parameters equal to one. The reason is that we want just to illustrate the procedure to reduce that system to only one ordinary differential equation, such that we know its analytical solution. This idea will be applied to study the relations between a system of three ordinary differential equations, and a couple of partial differential equations. Lotka-Volterra equations are solved numerically by a fourth-order predictor-corrector method, which is initialized by an improved Euler method with a rather small time step because it is only a second-order algorithm. Then, it is proposed a model with three species, defined by ordinary differential equations.

基于改进的广义S变换求取地层品质因子Q值

地层的品质因子Q值对衡量地震波传播过程中的能量衰减以及地震资料的处理、解释有重要意义。在实际生产中求取品质因子最实用的方法是频谱比法,但传统的频谱比法面临时窗选取等问题。本文利用改进的广义S变换时频特性及与傅里叶谱相联系的特性,提取地层上、下界面对应的瞬时频谱,并通过拟合振幅比与频率的关系得到地层的品质因子Q值。数值模拟及实际资料处理均证明了该方法的有效性。

基于广义S变换地震高分辨率处理方法的改进及在流花11-1油田的应用

相比较早的小波变换和傅氏变换,广义S变换具有更好的时频局部性,但也存在低频信息易损失、弱反射层易丢失等问题。对基于广义S变换的地震高分辨率处理方法进行了改进,提出了新的处理思路。这种基于广义S变换的高分辨率处理技术,在提取并补偿高频信号的同时,也对低频信号进行了有效的保持。该项技术在流花11-1油田取得了良好的应用效果。

铬天青S分光光度法测膨化食品中铝

目的：探讨铬天青S作显色剂测定膨化食品中铝的方法。方法：样品经消化后,在乙二胺-盐酸缓冲介质存在下,铝与铬天青S和聚乙二醇辛基苯醚及溴代十六烷基吡啶形成稳定蓝色四元体系,在一定浓度范围内,其吸光度与铝含量成正比。结果：在铝含量为0～0.1 mg/L的范围内,有良好的线性关系,平均相关系数r=0.9998,回收率在91.0%～95.2%之间,相对标准偏差为1.72%～2.22%。结论：该方法显色灵敏、稳定性好、准确、干扰少,适用于测量膨化食品中的铝。

海管S型初始铺设仿真算法

为准确模拟初始铺管作业中管道和起始缆的形态,创建一个逼真的深水铺管起重船铺管作业虚拟训练环境.针对S型铺管作业中的初始铺管作业,以Euler-Bernoulli梁理论为基础,对初始铺管管道和缆索进行静态分析,建立几何非线性微分方程,且对管道与缆索微分方程边界条件难以确定导致方程无法求解的问题,提出一种基于微分求积法的迭代方法,该方法能够准确地实现边界条件的确立,从而完成微分方程的求解.仿真和实验分析不同作业状态下管道与缆索的形态与内力变化,结果证明了初始铺管整体算法的准确性.该方法提高了微分方程求解精度且计算量少,易于程序实现,可用于海上铺管作业方案的工程预演,可行性分析和优化作业方案等.

基于时频分析的微震P波和S波到时联合拾取方法

微震信号到时的精确拾取是震源定位的重要前提,准确获取微震信号P波、S波到时具有重要理论意义。基于时频分析原理与到时拾取原理,提出了基于时频分析的下山比较法。该方法的时频分析原理通过语谱图、功率密度谱图、连续两次FIR带通滤波获取了背景噪声的位置和规律、P波和S波初至前后微震信号的频率、振幅、能量变化以及光滑且利于迭代平均值比较的波形。该方法的到时拾取原理通过将全子波振幅的数学期望设置为阈值,并遵循P波和S波功率大小、到达先后、波形重叠三大关系迭代比较微震信号子波振幅,从而获得P波精确到时和S波峰值到时。利用模型试验比较了该方法较改进长短时窗(STA/LTA)方法的优越性,并在工程实例中获得验证。结果表明:该方法对比改进STA/LTA方法,前者可同时拾取P波精确到时与S波峰值到时而后者只能拾取P波精确到时,前者P波平均时差、标准差分别为后者的6.18‰、3.98‰,前者单次拾取所需平均计算时间、标准差分别为后者的43.99%、10.54%,前者到时拾取失败比例为0,而后者为15.63%。

Beutler改良法用于探讨小鼠全血中GSH含量随龄变化的研究

将 Beutler改良法中的采血量标定由量体积进一步改良为称血重 ,并用于探讨小鼠全血中 GSH随龄变化的规律 .在完成小鼠寿命实验的基础上 ,待小鼠满 1 0月龄后在 3个不同的年龄期分别测定了 GSH含量 ,发现 GSH含量随龄下降后至一基本稳定值 ,该值较高的小鼠其寿命也相应较长 .结果表明 ,GSH随龄下降可能是小鼠衰老进程的标志之一 ,二者有一定的相关性 .此外 ,经两次改进的 Beutler法较原法更完善、更简捷。