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 appl...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.展开更多
In this work,the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus(COVID-19).The SITR mathema...In this work,the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus(COVID-19).The SITR mathematical model is divided into four classes using fractal parameters for COVID-19 dynamics,namely,susceptible(S),infected(I),treatment(T),and recovered(R).The main idea of the presented method is based on the matrix representations of the exponential functions and their derivatives using collocation points.To indicate the usefulness of this method,we employ it in some cases.For error analysis of the method,the residual of the solutions is reviewed.The reported examples show that the method is reasonably efficient and accurate.展开更多
A B-spline Interpolation Transport Solver(BITS) based on a collocation method is developed. It solves transport equations as a generalized interpolation problem, taking the first-order accuracy in time and the second-...A B-spline Interpolation Transport Solver(BITS) based on a collocation method is developed. It solves transport equations as a generalized interpolation problem, taking the first-order accuracy in time and the second-order accuracy in space along with a predictor–corrector or under-relaxation iteration method. Numerical tests show that BITS can solve one-dimensional transport equations for tokamak plasma more accurately without additional computation cost, compared to the finite difference method transport solver which is widely used in existing tokamak transport codes.展开更多
Finite element method (FEM) is an efficient numerical tool for the solution of partial differential equations (PDEs). It is one of the most general methods when compared to other numerical techniques. PDEs posed in a ...Finite element method (FEM) is an efficient numerical tool for the solution of partial differential equations (PDEs). It is one of the most general methods when compared to other numerical techniques. PDEs posed in a variational form over a given space, say a Hilbert space, are better numerically handled with the FEM. The FEM algorithm is used in various applications which includes fluid flow, heat transfer, acoustics, structural mechanics and dynamics, electric and magnetic field, etc. Thus, in this paper, the Finite Element Orthogonal Collocation Approach (FEOCA) is established for the approximate solution of Time Fractional Telegraph Equation (TFTE) with Mamadu-Njoseh polynomials as grid points corresponding to new basis functions constructed in the finite element space. The FEOCA is an elegant mixture of the Finite Element Method (FEM) and the Orthogonal Collocation Method (OCM). Two numerical examples are experimented on to verify the accuracy and rate of convergence of the method as compared with the theoretical results, and other methods in literature.展开更多
This study presents a comparative analysis of verb-noun collocation errors made by Chinese EFL learners with different language proficiency.Linguistic data from two sub-corpora of CLEC(Chinese Learner English Corpus,G...This study presents a comparative analysis of verb-noun collocation errors made by Chinese EFL learners with different language proficiency.Linguistic data from two sub-corpora of CLEC(Chinese Learner English Corpus,Gui&Yang,1999)consist of the compositions written freely by Chinese senior high school students(ST2)and the 3rd-and 4th-year English majors of Chinese universities(ST6).The study results show that four types of verb-noun collocation errors are frequently made due to incorrect L1 translation,misuse of relative synonyms,misuse of delexical verbs,and misuse of part of speech;the number of errors decreases with the development of language proficiency,especially errors by incorrect L1 translation and relative synonyms.And for both ST2 and ST6 EFL learners,errors resulted from incorrect L1 translation account for nearly a half with the same high-frequency words.It is also found that ST6 learners’competence in collocation and synonym identification doesn’t grow with the increase of their vocabulary,as is shown in the case of overusing delexical verbs in collocation,the same error made by ST2 learners,who have a significantly weaker command of delexical verbs.Implications of the findings are also discussed with the purpose of raising the awareness of collocation pedagogy and enhancing EFL learners’collocation competence.展开更多
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 un...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.展开更多
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 im...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?”展开更多
The aim of this paper is to obtain the numerical solutions of generalized space-fractional Burgers' equations with initial-boundary conditions by the Jacobi spectral collocation method using the shifted Jacobi-Gau...The aim of this paper is to obtain the numerical solutions of generalized space-fractional Burgers' equations with initial-boundary conditions by the Jacobi spectral collocation method using the shifted Jacobi-Gauss-Lobatto collocation points. By means of the simplifed Jacobi operational matrix, we produce the diferentiation matrix and transfer the space-fractional Burgers' equation into a system of ordinary diferential equations that can be solved by the fourth-order Runge-Kutta method. The numerical simulations indicate that the Jacobi spectral collocation method is highly accurate and fast convergent for the generalized space-fractional Burgers' equation.展开更多
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 college...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.展开更多
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 nativ...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.展开更多
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'...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.展开更多
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...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.展开更多
In this paper,vibration analysis of functionally graded porous beams is carried out using the third-order shear deformation theory.The beams have uniform and non-uniform porosity distributions across their thickness a...In this paper,vibration analysis of functionally graded porous beams is carried out using the third-order shear deformation theory.The beams have uniform and non-uniform porosity distributions across their thickness and both ends are supported by rotational and translational springs.The material properties of the beams such as elastic moduli and mass density can be related to the porosity and mass coefficient utilizing the typical mechanical features of open-cell metal foams.The Chebyshev collocation method is applied to solve the governing equations derived from Hamilton's principle,which is used in order to obtain the accurate natural frequencies for the vibration problem of beams with various general and elastic boundary conditions.Based on the numerical experiments,it is revealed that the natural frequencies of the beams with asymmetric and non-uniform porosity distributions are higher than those of other beams with uniform and symmetric porosity distributions.展开更多
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 renew...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.展开更多
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 re...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.展开更多
基金the National Natural Science Foundation of China for financial support to this work under Grant NSFC No.12072064.
文摘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.
文摘In this work,the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus(COVID-19).The SITR mathematical model is divided into four classes using fractal parameters for COVID-19 dynamics,namely,susceptible(S),infected(I),treatment(T),and recovered(R).The main idea of the presented method is based on the matrix representations of the exponential functions and their derivatives using collocation points.To indicate the usefulness of this method,we employ it in some cases.For error analysis of the method,the residual of the solutions is reviewed.The reported examples show that the method is reasonably efficient and accurate.
基金the National MCF Energy R&D Program of China(No.2019YFE03040004)the Comprehensive Research Facility for Fusion Technology Program of China(No.2018-000052-73-01-001228)the National MCF Energy R&D Program of China(No.2019YFE03060000)。
文摘A B-spline Interpolation Transport Solver(BITS) based on a collocation method is developed. It solves transport equations as a generalized interpolation problem, taking the first-order accuracy in time and the second-order accuracy in space along with a predictor–corrector or under-relaxation iteration method. Numerical tests show that BITS can solve one-dimensional transport equations for tokamak plasma more accurately without additional computation cost, compared to the finite difference method transport solver which is widely used in existing tokamak transport codes.
文摘Finite element method (FEM) is an efficient numerical tool for the solution of partial differential equations (PDEs). It is one of the most general methods when compared to other numerical techniques. PDEs posed in a variational form over a given space, say a Hilbert space, are better numerically handled with the FEM. The FEM algorithm is used in various applications which includes fluid flow, heat transfer, acoustics, structural mechanics and dynamics, electric and magnetic field, etc. Thus, in this paper, the Finite Element Orthogonal Collocation Approach (FEOCA) is established for the approximate solution of Time Fractional Telegraph Equation (TFTE) with Mamadu-Njoseh polynomials as grid points corresponding to new basis functions constructed in the finite element space. The FEOCA is an elegant mixture of the Finite Element Method (FEM) and the Orthogonal Collocation Method (OCM). Two numerical examples are experimented on to verify the accuracy and rate of convergence of the method as compared with the theoretical results, and other methods in literature.
文摘This study presents a comparative analysis of verb-noun collocation errors made by Chinese EFL learners with different language proficiency.Linguistic data from two sub-corpora of CLEC(Chinese Learner English Corpus,Gui&Yang,1999)consist of the compositions written freely by Chinese senior high school students(ST2)and the 3rd-and 4th-year English majors of Chinese universities(ST6).The study results show that four types of verb-noun collocation errors are frequently made due to incorrect L1 translation,misuse of relative synonyms,misuse of delexical verbs,and misuse of part of speech;the number of errors decreases with the development of language proficiency,especially errors by incorrect L1 translation and relative synonyms.And for both ST2 and ST6 EFL learners,errors resulted from incorrect L1 translation account for nearly a half with the same high-frequency words.It is also found that ST6 learners’competence in collocation and synonym identification doesn’t grow with the increase of their vocabulary,as is shown in the case of overusing delexical verbs in collocation,the same error made by ST2 learners,who have a significantly weaker command of delexical verbs.Implications of the findings are also discussed with the purpose of raising the awareness of collocation pedagogy and enhancing EFL learners’collocation competence.
基金NNSF of China, SF of SEC of China and SF of Wuhan University.
文摘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.
文摘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?”
基金This work is supported by the National Natural Science Foundation of China(Grant Nos.11701358,11774218)。
文摘The aim of this paper is to obtain the numerical solutions of generalized space-fractional Burgers' equations with initial-boundary conditions by the Jacobi spectral collocation method using the shifted Jacobi-Gauss-Lobatto collocation points. By means of the simplifed Jacobi operational matrix, we produce the diferentiation matrix and transfer the space-fractional Burgers' equation into a system of ordinary diferential equations that can be solved by the fourth-order Runge-Kutta method. The numerical simulations indicate that the Jacobi spectral collocation method is highly accurate and fast convergent for the generalized space-fractional Burgers' equation.
文摘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.
文摘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.
文摘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.
文摘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.
文摘In this paper,vibration analysis of functionally graded porous beams is carried out using the third-order shear deformation theory.The beams have uniform and non-uniform porosity distributions across their thickness and both ends are supported by rotational and translational springs.The material properties of the beams such as elastic moduli and mass density can be related to the porosity and mass coefficient utilizing the typical mechanical features of open-cell metal foams.The Chebyshev collocation method is applied to solve the governing equations derived from Hamilton's principle,which is used in order to obtain the accurate natural frequencies for the vibration problem of beams with various general and elastic boundary conditions.Based on the numerical experiments,it is revealed that the natural frequencies of the beams with asymmetric and non-uniform porosity distributions are higher than those of other beams with uniform and symmetric porosity distributions.
基金supported by the National Natural Science Foundation of China (NSFC) (No. 51577075)
文摘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.
基金The project supported by the National Natural Science Foundation of China (10172052)
文摘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.