Review of Collocation Methods and Applications in Solving Science and Engineering Problems

The collocation method is a widely used numerical method for science and engineering problems governed by partial differential equations.This paper provides a comprehensive review of collocation methods and their applications,focused on elasticity,heat conduction,electromagnetic field analysis,and fluid dynamics.The merits of the collocation method can be attributed to the need for element mesh,simple implementation,high computational efficiency,and ease in handling irregular domain problems since the collocation method is a type of node-based numerical method.Beginning with the fundamental principles of the collocation method,the discretization process in the continuous domain is elucidated,and how the collocation method approximation solutions for solving differential equations are explained.Delving into the historical development of the collocation methods,their earliest applications and key milestones are traced,thereby demonstrating their evolution within the realm of numerical computation.The mathematical foundations of collocation methods,encompassing the selection of interpolation functions,definition of weighting functions,and derivation of integration rules,are examined in detail,emphasizing their significance in comprehending the method’s effectiveness and stability.At last,the practical application of the collocation methods in engineering contexts is emphasized,including heat conduction simulations,electromagnetic coupled field analysis,and fluid dynamics simulations.These specific case studies can underscore collocation method’s broad applicability and effectiveness in addressing complex engineering challenges.In conclusion,this paper puts forward the future development trend of the collocation method through rigorous analysis and discussion,thereby facilitating further advancements in research and practical applications within these fields.

A Collocation Technique via Pell-Lucas Polynomials to Solve Fractional Differential EquationModel for HIV/AIDS with Treatment Compartment

In this study,a numerical method based on the Pell-Lucas polynomials(PLPs)is developed to solve the fractional order HIV/AIDS epidemic model with a treatment compartment.The HIV/AIDS mathematical model with a treatment compartment is divided into five classes,namely,susceptible patients(S),HIV-positive individuals(I),individuals with full-blown AIDS but not receiving ARV treatment(A),individuals being treated(T),and individuals who have changed their sexual habits sufficiently(R).According to the method,by utilizing the PLPs and the collocation points,we convert the fractional order HIV/AIDS epidemic model with a treatment compartment into a nonlinear system of the algebraic equations.Also,the error analysis is presented for the Pell-Lucas approximation method.The aim of this study is to observe the behavior of five populations after 200 days when drug treatment is applied to HIV-infectious and full-blown AIDS people.To demonstrate the usefulness of this method,the applications are made on the numerical example with the help of MATLAB.In addition,four cases of the fractional order derivative(p=1,p=0.95,p=0.9,p=0.85)are examined in the range[0,200].Owing to applications,we figured out that the outcomes have quite decent errors.Also,we understand that the errors decrease when the value of N increases.The figures in this study are created in MATLAB.The outcomes indicate that the presented method is reasonably sufficient and correct.

Effects of Collocational Congruency,Frequency and L2 Proficiency on the Processing of English Adjective-Noun Collocations

The present study investigates the effects of congruency and frequency on adjective-noun collocational processing for Chinese learners of English at two proficiency levels based on the data obtained in an online acceptability judgment task.The subject pool of this research included 60 English majors studying at a university in China;30 were selected as a higher-proficiency group and 30 as a lower-proficiency group according to their Vocabulary Levels Test(Schmitt et al.,2001)scores and their self-reported proficiency in English.The experimental materials were programmed to E-prime 2.0 and included six types of collocations:(1)15 high-frequency congruent collocations,(2)15 low-frequency congruent collocations,(3)15 high-frequency incongruent collocations,(4)15 low-frequency incongruent collocations,(5)15 Chinese-only items,and(6)75 unrelated items for baseline data.The collected response times(RTs)and accuracy rates data were statistically analyzed by the use of an ANOVA test and pairwise comparisons through SPSS 16.0 software.The results revealed that:(1)the adjective-noun collocational processing of Chinese English learners is influenced by collocational frequency,congruency and L2 proficiency;(2)the processing time is affected by the interaction of congruency and frequency;and(3)the interactive effect of L2 proficiency in conjunction with congruency and frequency also influences the processing quality.

基于Triple Collocation方法的土壤湿度误差分析

土壤湿度作为水循环中重要的驱动参数之一,对天气变化有着显著影响。遥感技术的发展促使土壤湿度的大范围动态性观测变为可能,但对其误差的准确估计仍需进一步的研究。利用ASCAT散射计、AMSR-E辐射计反演得到的2种卫星遥感土壤湿度数据以及ERA-Interim土壤湿度再分析资料,通过三重组合(Triple Collocation,TC)方法得到了研究区域(15°N^55°N,73°E^135°E) 3种土壤湿度数据的误差方差和信噪比估计,并结合MODIS土地覆盖类型数据分析了3种土壤湿度数据的误差特征。结果表明:植被覆盖会影响遥感土壤湿度的TC误差方差估计;从TC误差方差估计值来看,ERA土壤湿度精度最高,AMSR-E精度次之,ASCAT精度最低;从信噪比来看,ASCAT土壤湿度信噪比最高,ERA的信噪比低于ASCAT高于AMSR-E,AMSR-E信噪比最低;通过研究区MODIS土地覆盖类型数据与TC结果的分析可知TC结果多分布在草原、农田和裸地,TC结果比较符合客观实际情况。

小步长低阶Collocation方法轨道积分性能研究

介绍了Collocation方法的原理及误差构成,以Collocation方法在逐历元滤波轨道解算中的应用为切入点对小步长低阶Collocation方法轨道积分的性能进行了研究。分别就二体问题及真实GPS星座卫星轨道为对象进行了一系列验算,得到了一些有益结论:就积分方法本身性能而言,Collocation积分方法较传统数值积分法精度要高且稳定性要好;在一定长度的积分步长范围内,将小步长低阶Collocation方法用于滤波轨道解算是可行且有效的。但在积分步长很小的情形下,从计算效率方面考虑,不推荐使用Collocation方法;利用小步长低阶Collocation方法取代大步长高阶方法能提高效率且可抑制蚀现象引起的积分结果发散。经试验表明:选用积分步长为900s的4阶方法是合适的。

Collocation-PECE算法在低轨卫星精密定轨中的应用

在精密轨道确定中,PECE算法已成为目前最通用的轨道积分算法。利用Collocation方法的外推公式和内插公式可构造一种新的PECE算法,新算法以两种方式进行二阶微分方程的求解:将二阶微分方程降为一阶微分方程然后积分以及直接积分二阶微分方程。分别采用这两种类型的算法求解二体问题和低轨卫星的受摄轨道,并将计算结果与Adams-PECE算法及一般的PECE算法的积分结果进行了比较分析。结果表明,这两种类型的Collo-cation-PECE算法在实际轨道积分中较其它几种算法更有效。

基于统计的常用词搭配(Collocation)的发现方法

常用词搭配 (collocation)是指一些常用的用来表达某些事情或事物的短语 ,一般是二元组或三元组等。常用词搭配的自动发现在自然语言处理中起着很重要的作用 ,它能够丰富词典的容量 ,提高系统的性能。本文提出 4种判断一个二元组是否是常用词搭配的方法 ,并对各种方法的结果进行了比较。然后在已知二元组的基础上 ,提出了统计和发现三元或多元组的方法。这种利用二元组来构造三元组的方法 ,比统计所有三元组的计算量大大减少。

Idioms,Collocations,Formulaic Phrases and Second Language Acquisition

Idioms,collocations,and formulaic phrases are three factors playing the important role in the second language acquisition(SLA).This essay firstly deals with the definitions of these concepts,then it illustrates the importance of them to SLA on two perspectives—idioms,collocations,and formulaic phrases respectively and they regarded as a whole.Therefore,in this way,the essay may lead to the answer the question“How are idioms,collocations,and formulaic phrases important to SLA?”

一种改进的collocation方法及其在动态定价问题中的应用

涉及多个状态变量的动态定价问题往往难以求解。基于此,引入多维插值与张量积,提出一种改进的collocation方法用于求解多状态变量的动态优化问题,并从理论上分析了改进后的collocation方法的收敛性。首先,将改进的collocation方法应用于求解多个状态变量的动态经济问题;给出了改进后的collocation方法的实际运用,应用于解决有限记忆的动态定价问题;利用状态变量的切比雪夫格点与其估计值构造出时序的状态转移,并依据状态转移模拟最优价格路径。通过数值分析表明,改进后的collocation方法比线性二次逼近法更快的收敛于稳态。

Study on the Collocation between Small Garden Ornaments and Other Elements

Starting with the basic elements of small garden ornaments, this research discusses its collocation art with other landscape elements such as plants, water, stones and environment, in order to offer useful references for the garden planning in a city.

THE ACCURACY COMPARISON BETWEEN CHEBYSHEV-τ METHOD AND CHEBYSHEV COLLOCATION METHOD

This paper is devoted to investigate the accuracy of the Pseudo spectral scheme with the Chebyshev tau method and Chebyshev collocation method. The computational results of the nonlinear disturbance development in plane Poiseuille flow for both methods are presented and compared in detail. It is acknowledged that the Chebyshev collocation method has higher precision than the other one, especially for near netural situation.

基于collocation方法的Taylor-Culick模型流动稳定性的特征向量分析

对于固体火箭发动机中产生的流动不稳定现象,需要利用简化模型进行分析,目前最符合实际的简化模型为Taylor-Culick流模型。采用稳定性理论对轴对称的Taylor-Culick模型进行数值计算时,由于对称轴处径向坐标为零而带来奇性,以往的数值计算方法难以处理。采用一种配置微分矩阵的方式来改进算法,避免柱坐标所带有的奇性对数值计算的影响。对固体火箭发动机Taylor-Culick流动模型进行了局部稳定性计算,得到了一致的结果。分析了特征向量随计算参数的变化,发现频率、雷诺数与特征向量的形态变化有直接关系,且特征向量的变化代表着特定情况下流动区域振荡发生的范围与形式,这些流动不稳定的局部特征从细节上反映了整体流动对应于不同流动参数的状态。

Chinese EFL Teachers' Attitudes toward Corpus Use in Collocation Instruction

The purpose of the present study was to investigate Chinese EFL(English as a Foreign Language)teachers’attitudes toward corpus use in collocation instruction.Fourteen Chinese EFL teachers from seven different colleges or universities answered a questionnaire asking their perceptions about corpus use in collocation instruction.The statistical analysis revealed that significant difference(t=2.449,df.=9,p=.037<0.1)was found between males and females in their perception of the learnability of the searching technique in collocation corpus.Besides,the result also revealed that significant differences were found between novice and experienced teachers’perception of helpfulness of corpus in writing native-like sentences(t=5.75,df=11,p<0.1),identifying collocation errors in English(t=5.75,df=11,p<0.1),usefulness of corpus in searching for English collocations(t=2.93,df=12,p<0.1),and advocacy of corpus to colleagues(t=5.75,df=11,p<0.1).Finally,some pedagogical implications were put forward.

A Corpus-based Analysis of Adjective/Noun Collocationwith Big and Large as Examples

Adopting the corpus-based approach, this paper analyzes the adjective/noun collocation with two English words big and large as examples by comparing data from Chinese Learner English Corpus (CLEC) and those from native speaker corpora British National Corpus (BNC). It is observed that collocates chosen by Chinese students differ greatly from those chosen by native speakers, on the basis of which this paper compares and illustrates the differences of these words and aims to highlight the important role of the corpus-based collocational research in English vocabulary teaching.

On the Grammatical Unaccountability of English Lexical Collocation from the Perspective of Pragmatics

As for the English lexical collocation,there has been a great amount of researches conducted by Chinese and foreign scholars.However,less people has offered the explanation in allusion to the collocation which isn't confined to the grammatical domain.Therefore,the thesis aims to depart from the research of the grammatical unaccountability of the lexical collocation to deeply investigate the reason of it from the point view of etymology,semantics,pragmatics and cultural category by taking"female secretary" as an example.In addition,the thesis intends to search for an effective learning way to help learners improve their learning efficiency and use the English lexicon correctly as a native speaker.

Collocation的语篇衔接功能

人们对collocation的认识已由语义层次的“习惯搭配”发展到语篇层次的“词汇同现”。collocation在语篇结构中的衔接作用:通过语义联想使词汇同现,相互联系,形成词汇套,具有连句成篇的功能;通过语义场使词汇同现,形成以“场”为纽带的词汇套,具有连句成篇的功能。

THE COLLOCATION METHODS FOR SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNELS

This paper applies the singular integral operators, singular quadrature operators and discretization matrices associated with singular integral equations with Cauchy kernels, which are established in [1], to give a unified framework for various collocation methods of numerical solutions of singular integral equations with Cauchy kernels. Under the framework, the coincidence of the direct quadrature method and the indirect quadrature method is very simple and obvious.

A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate

In this paper,a deep collocation method(DCM)for thin plate bending problems is proposed.This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning.Besides,the proposed DCM is based on a feedforward deep neural network(DNN)and differs from most previous applications of deep learning for mechanical problems.First,batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries.A loss function is built with the aim that the governing partial differential equations(PDEs)of Kirchhoff plate bending problems,and the boundary/initial conditions are minimised at those collocation points.A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters.In Kirchhoff plate bending problems,the C^1 continuity requirement poses significant difficulties in traditional mesh-based methods.This can be solved by the proposed DCM,which uses a deep neural network to approximate the continuous transversal deflection,and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.

MESHLESS METHOD BASED ON COLLOCATION WITH CONSISTENT COMPACTLY SUPPORTED RADIAL BASIS FUNCTIONS

Based on our previous study,the accuracy of derivatives of interpolating functions are usually very poor near the boundary of domain when Compactly Supported Radial Basis Functions (CSRBFs)are used,so that it could result in significant error in solving partial differential equations with Neumann boundary conditions.To overcome this drawback,the Consistent Compactly Supported Radial Basis Functions(CCSRBFs)are developed,which satisfy the predetermined consistency con- ditions.Meshless method based on point collocation with CCSRBFs is developed for solving partial differential equations.Numerical studies show that the proposed method improves the accuracy of approximation significantly.

Probabilistic small signal stability analysis of power system with wind power and photovoltaic power based on probability collocation method

Recently, with increasing improvements in the penetration of wind power and photovoltaic power in the world, probabilistic small signal stability analysis(PSSSA) of a power system consisting of multiple types of renewable energy has become a key problem. To address this problem, this study proposes a probabilistic collocation method(PCM)-based PSSSA for a power system consisting of wind farms and photovoltaic farms. Compared with the conventional Monte Carlo method, the proposed method meets the accuracy and precision requirements and greatly reduces the computation; therefore, it is suitable for the PSSSA of this power system. Case studies are conducted based on a 4-machine 2-area and New England systems, respectively. The simulation results show that, by reducing synchronous generator output to improve the penetration of renewable energy, the probabilistic small signal stability(PSSS) of the system is enhanced. Conversely, by removing part of the synchronous generators to improve the penetration of renewable energy, the PSSS of the system may be either enhanced or deteriorated.