In [1], a class of multiderivative block methods (MDBM) was studied for the numerical solutions of stiff ordinary differential equations. This paper is aimed at solving the problem proposed in [1] that what conditions...In [1], a class of multiderivative block methods (MDBM) was studied for the numerical solutions of stiff ordinary differential equations. This paper is aimed at solving the problem proposed in [1] that what conditions should be fulfilled for MDBMs in order to guarantee the A-stabilities. The explicit expressions of the polynomialsP(h) and Q(h) in the stability functions h(h)=P(h)/Q(h)are given. Furthermore, we prove P(-h)-Q(h). With the aid of symbolic computations and the expressions of diagonal Fade approximations, we obtained the biggest block size k of the A-stable MDBM for any given l (the order of the highest derivatives used in MDBM,l>1)展开更多
Many initial value problems are difficult to be solved using ordinary,explicit step-by-step methods because most of these problems are considered stiff.Certain implicit methods,however,are capable of solving stiff ord...Many initial value problems are difficult to be solved using ordinary,explicit step-by-step methods because most of these problems are considered stiff.Certain implicit methods,however,are capable of solving stiff ordinary differential equations(ODEs)usually found in most applied problems.This study aims to develop a new numerical method,namely the high order variable step variable order block backward differentiation formula(VSVOHOBBDF)for the main purpose of approximating the solutions of third order ODEs.The computational work of the VSVO-HOBBDF method was carried out using the strategy of varying the step size and order in a single code.The order of the proposed method was then discussed in detail.The advancement of this strategy is intended to enhance the efficiency of the proposed method to approximate solutions effectively.In order to confirm the efficiency of the VSVO-HOBBDF method over the two ODE solvers in MATLAB,particularly ode15s and ode23s,a numerical experiment was conducted on a set of stiff problems.The numerical results prove that for this particular set of problem,the use of the proposed method is more efficient than the comparable methods.VSVO-HOBBDF method is thus recommended as a reliable alternative solver for the third order ODEs.展开更多
In this paper, the existence and uniqueness of the solution of Fredholm-Volterra integral equation is considered (NF-VIE) with continuous kernel;then we used a numerical method to reduce this type of equations to a sy...In this paper, the existence and uniqueness of the solution of Fredholm-Volterra integral equation is considered (NF-VIE) with continuous kernel;then we used a numerical method to reduce this type of equations to a system of nonlinear Volterra integral equations. Runge-Kutta method (RKM) and Bolck by block method (BBM) are used to solve the system of nonlinear Volterra integral equations of the second kind (SNVIEs) with continuous kernel. The error in each case is calculated.展开更多
In this paper, we provide a generalized block-by-block method for constructing block-by-block systems to solve the system of linear Volterra integral equations of the second kind, and then deduce some of the special c...In this paper, we provide a generalized block-by-block method for constructing block-by-block systems to solve the system of linear Volterra integral equations of the second kind, and then deduce some of the special cases. Compared with the expansion method and He＇s homotopy perturbation method, respectively numerical examples are given to certify the effectiveness of the method. The results show that the block-by-block method is very effective, simple, and of high accuracy in solving the system of linear Volterra integral equations of the second kind.展开更多
In this paper, a block method with one hybrid point for solving Jerk equations is presented. The hybrid point is chosen to optimize the local truncation errors of the main formulas for the solution and the derivative ...In this paper, a block method with one hybrid point for solving Jerk equations is presented. The hybrid point is chosen to optimize the local truncation errors of the main formulas for the solution and the derivative at the end of the block. Analysis of the method is discussed, and some numerical examples show that the proposed method is efficient and accurate.展开更多
In this paper, we developed a new continuous block method using the approach of collocation of the differential system and interpolation of the power series approximate solution. A constant step length within a half s...In this paper, we developed a new continuous block method using the approach of collocation of the differential system and interpolation of the power series approximate solution. A constant step length within a half step interval of integration was adopted. We evaluated at grid and off grid points to get a continuous linear multistep method. The continuous linear multistep method is solved for the independent solution to yield a continuous block method which is evaluated at selected points to yield a discrete block method. The basic properties of the block method were investigated and found to be consistent and zero stable hence convergent. The new method was tested on real life problems namely: SIR model, Growth model and Mixture Model. The results were found to compete favorably with the existing methods in terms of accuracy and error bound.展开更多
In this paper, we developed a new continuous block method by the method of interpolation and collocation to derive new scheme. We adopted the use of power series as a basis function for approximate solution. We evalua...In this paper, we developed a new continuous block method by the method of interpolation and collocation to derive new scheme. We adopted the use of power series as a basis function for approximate solution. We evaluated at off grid points to get a continuous hybrid multistep method. The continuous hybrid multistep method is solved for the independent solution to yield a continuous block method which is evaluated at selected points to yield a discrete block method. The basic properties of the block method were investigated and found to be consistent, zero stable and convergent. The results were found to compete favorably with the existing methods in terms of accuracy and error bound. In particular, the scheme was found to have a large region of absolute stability. The new method was tested on real life problem namely: Dynamic model.展开更多
Neutral Delay Differential Equation(NDDE)is a differential problem that has regularly existed in numerous occurrences and has represented a significant role in dealing with real-life phenomena,especially on their appl...Neutral Delay Differential Equation(NDDE)is a differential problem that has regularly existed in numerous occurrences and has represented a significant role in dealing with real-life phenomena,especially on their application in biological and physiological processes.A fifth-order two-point hybrid implicit multistep block method(2PIH5)has been formulated in this research for the numerical solution of Neutral Delay Differential Equation(NDDE).A Taylor series interpolation polynomial has been implemented in the formulation of the proposed 2PIH5.The order,consistency,and zero-stability for 2PIH5 have been illustrated.The analyses of convergence and stability test have been performed and discussed.The initial value problems for the first-order NDDE with constant or proportional delay have been solved using the proposed block method.Some numerical results for the proposed method have been presented to prove the adaptability and applicability of the proposed method in solving NDDE.The proposed method is proved to be comparable with the other existing methods.It is assumed to be reliable and efficient for solving the first-order NDDE with constant or proportional delay.展开更多
This paper focuses on the numerical stability of the block θ methods adapted to differential equations with a delay argument. For the block θ methods, an interpolation procedure is introduced which leads to the nume...This paper focuses on the numerical stability of the block θ methods adapted to differential equations with a delay argument. For the block θ methods, an interpolation procedure is introduced which leads to the numerical processes that satisfy an important asymptotic stability condition related to the class of test problems y′(t)=ay(t)+by(t-τ) with a,b∈C, Re(a)<-|b| and τ>0. We prove that the block θ method is GP stable if and only if the method is A stable for ordinary differential equations. Furthermore, it is proved that the P and GP stability are equivalent for the block θ method.展开更多
The symmetric linear system gives us many simplifications and a possibility to adapt the computations to the computer at hand in order to achieve better performance. The aim of this paper is to consider the block bidi...The symmetric linear system gives us many simplifications and a possibility to adapt the computations to the computer at hand in order to achieve better performance. The aim of this paper is to consider the block bidiagonalization methods derived from a symmetric augmented multiple linear systems and make a comparison with the block GMRES and block biconjugate gradient methods.展开更多
Based on the limit analysis upper bound method, a new mechanism of soil slope failure has been proposed which was consisted of plastic shear zone and rigid block zone. The different zones interface were regarded as di...Based on the limit analysis upper bound method, a new mechanism of soil slope failure has been proposed which was consisted of plastic shear zone and rigid block zone. The different zones interface were regarded as discontinuity lines. Two sliding blocks of the slope were also incorporated horizontal seismic force and vertical gravity force. The velocities and forces were analyzed in two blocks, and the expression of velocity discontinuities was derived according to the principle of incompressibility. The external force done work for the blocks and the internal energy dissipated of the plastic shear zone and the velocity discontinuous were solved.The stability ratios were derived for the height of two-level slope with different rates to involve seismic and no seismic. The present stability ratios were compared to the previous study, which showed the superiority of the mechanism and the rationality of the analysis. The critical height of the slope can provide a theoretical basis for slope support and design.展开更多
Based on elementary group theory, the block pivot methods for solving two-dimensional elastic frictional contact problems are presented in this paper. It is proved that the algorithms converge within a finite number o...Based on elementary group theory, the block pivot methods for solving two-dimensional elastic frictional contact problems are presented in this paper. It is proved that the algorithms converge within a finite number of steps when the friction coefficient is ''relative small''. Unlike most mathematical programming methods for contact problems, the block pivot methods permit multiple exchanges of basic and nonbasic variables.展开更多
This paper presents a class of r-point (r+1)st-order A-stable one-block methods with damping at the infinite point (DIAOB r,r+1 ). Under the conditions of the same order, A-stability, operation count (at each iterativ...This paper presents a class of r-point (r+1)st-order A-stable one-block methods with damping at the infinite point (DIAOB r,r+1 ). Under the conditions of the same order, A-stability, operation count (at each iterative step) and storage space are the same as the methods in , the methods in the paper improve the stability in a neighborhood at the infinite point. And, by using the OOPI method , it possesses much faster rate of convergence for solving systems of nonlinear equations produced by the DIAOB r,r+1 .展开更多
Iterative methods that take advantage of efficient block operations and block communications are popular research topics in parallel computation. These methods are especially important on Massively Parallel Processors...Iterative methods that take advantage of efficient block operations and block communications are popular research topics in parallel computation. These methods are especially important on Massively Parallel Processors (MPP). This paper presents a block variant of the GMRES method for solving general unsymmetric linear systems. It is shown that the new algorithm with block size s, denoted by BVGMRES(s,m), is theoretically equivalent to the GMRES(s. m) method. The numerical results show that this algorithm can be more efficient than the standard GMRES method on a cache based single CPU computer with optimized BLAS kernels. Furthermore, the gain in efficiency is more significant on MPPs due to both efficient block operations and efficient block data communications. Our numerical results also show that in comparison to the standard GMRES method, the more PEs that are used on an MPP, the more efficient the BVGMRES(s,m) algorithm is.展开更多
Text extraction is an important initial step in digitizing the historical documents. In this paper, we present a text extraction method for historical Tibetan document images based on block projections. The task of te...Text extraction is an important initial step in digitizing the historical documents. In this paper, we present a text extraction method for historical Tibetan document images based on block projections. The task of text extraction is considered as text area detection and location problem. The images are divided equally into blocks and the blocks are filtered by the information of the categories of connected components and corner point density. By analyzing the filtered blocks' projections, the approximate text areas can be located, and the text regions are extracted. Experiments on the dataset of historical Tibetan documents demonstrate the effectiveness of the proposed method.展开更多
OBJECTIVE: To compare the clinical effect of brachial plexus block with "One Injection Two Points" guided under ultrasound and the conventional method guiding by ultrasound. METHODS: 70 patients were randomi...OBJECTIVE: To compare the clinical effect of brachial plexus block with "One Injection Two Points" guided under ultrasound and the conventional method guiding by ultrasound. METHODS: 70 patients were randomized evenly into 2 groups, with 35 patients in each group, while the Experiment Group(Group B) received One Injection Two Points" method, the Control Group(Group A) received the conventional method.The nerve block every 5 s, the success rate of anesthesia, the dosage of local anesthetics, second remedial anesthesia, adverse reactions, etc.were recorded. RESULTS: Group B was superior to group A in the success rate of anesthesia; There were 6 patients in group A who required constant pump injection of Remifentanil to remedy, while no patients in Group B needed remedy treatment. There were no serious adverse reactions in both groups.CONCLUSIONS: The clinical effect of brachial plexus block with "One Injection Two Points" method guided under ultrasoundguiding by ultrasound was superior to that of the conventional method.展开更多
文摘In [1], a class of multiderivative block methods (MDBM) was studied for the numerical solutions of stiff ordinary differential equations. This paper is aimed at solving the problem proposed in [1] that what conditions should be fulfilled for MDBMs in order to guarantee the A-stabilities. The explicit expressions of the polynomialsP(h) and Q(h) in the stability functions h(h)=P(h)/Q(h)are given. Furthermore, we prove P(-h)-Q(h). With the aid of symbolic computations and the expressions of diagonal Fade approximations, we obtained the biggest block size k of the A-stable MDBM for any given l (the order of the highest derivatives used in MDBM,l>1)
基金funded by Fundamental Research Grant Scheme Universiti Sains Malaysia,Grant No.203/PJJAUH/6711688 received by S.A.M.Yatim.Url at http://www.research.usm.my/default.asp?tag=3&f=1&k=1.
文摘Many initial value problems are difficult to be solved using ordinary,explicit step-by-step methods because most of these problems are considered stiff.Certain implicit methods,however,are capable of solving stiff ordinary differential equations(ODEs)usually found in most applied problems.This study aims to develop a new numerical method,namely the high order variable step variable order block backward differentiation formula(VSVOHOBBDF)for the main purpose of approximating the solutions of third order ODEs.The computational work of the VSVO-HOBBDF method was carried out using the strategy of varying the step size and order in a single code.The order of the proposed method was then discussed in detail.The advancement of this strategy is intended to enhance the efficiency of the proposed method to approximate solutions effectively.In order to confirm the efficiency of the VSVO-HOBBDF method over the two ODE solvers in MATLAB,particularly ode15s and ode23s,a numerical experiment was conducted on a set of stiff problems.The numerical results prove that for this particular set of problem,the use of the proposed method is more efficient than the comparable methods.VSVO-HOBBDF method is thus recommended as a reliable alternative solver for the third order ODEs.
文摘In this paper, the existence and uniqueness of the solution of Fredholm-Volterra integral equation is considered (NF-VIE) with continuous kernel;then we used a numerical method to reduce this type of equations to a system of nonlinear Volterra integral equations. Runge-Kutta method (RKM) and Bolck by block method (BBM) are used to solve the system of nonlinear Volterra integral equations of the second kind (SNVIEs) with continuous kernel. The error in each case is calculated.
基金Supported by the National Natural Science Foundation of China(10962008)
文摘In this paper, we provide a generalized block-by-block method for constructing block-by-block systems to solve the system of linear Volterra integral equations of the second kind, and then deduce some of the special cases. Compared with the expansion method and He＇s homotopy perturbation method, respectively numerical examples are given to certify the effectiveness of the method. The results show that the block-by-block method is very effective, simple, and of high accuracy in solving the system of linear Volterra integral equations of the second kind.
文摘In this paper, a block method with one hybrid point for solving Jerk equations is presented. The hybrid point is chosen to optimize the local truncation errors of the main formulas for the solution and the derivative at the end of the block. Analysis of the method is discussed, and some numerical examples show that the proposed method is efficient and accurate.
文摘In this paper, we developed a new continuous block method using the approach of collocation of the differential system and interpolation of the power series approximate solution. A constant step length within a half step interval of integration was adopted. We evaluated at grid and off grid points to get a continuous linear multistep method. The continuous linear multistep method is solved for the independent solution to yield a continuous block method which is evaluated at selected points to yield a discrete block method. The basic properties of the block method were investigated and found to be consistent and zero stable hence convergent. The new method was tested on real life problems namely: SIR model, Growth model and Mixture Model. The results were found to compete favorably with the existing methods in terms of accuracy and error bound.
文摘In this paper, we developed a new continuous block method by the method of interpolation and collocation to derive new scheme. We adopted the use of power series as a basis function for approximate solution. We evaluated at off grid points to get a continuous hybrid multistep method. The continuous hybrid multistep method is solved for the independent solution to yield a continuous block method which is evaluated at selected points to yield a discrete block method. The basic properties of the block method were investigated and found to be consistent, zero stable and convergent. The results were found to compete favorably with the existing methods in terms of accuracy and error bound. In particular, the scheme was found to have a large region of absolute stability. The new method was tested on real life problem namely: Dynamic model.
基金All authors gratefully acknowledge for the financial support by Putra Grant(project code:GP-IPS/2018/9625400)Graduate Research Fellowship(GRF)from Universiti Putra Malaysia.The authors are also thankful to the referees for their useful comments and suggestions.
文摘Neutral Delay Differential Equation(NDDE)is a differential problem that has regularly existed in numerous occurrences and has represented a significant role in dealing with real-life phenomena,especially on their application in biological and physiological processes.A fifth-order two-point hybrid implicit multistep block method(2PIH5)has been formulated in this research for the numerical solution of Neutral Delay Differential Equation(NDDE).A Taylor series interpolation polynomial has been implemented in the formulation of the proposed 2PIH5.The order,consistency,and zero-stability for 2PIH5 have been illustrated.The analyses of convergence and stability test have been performed and discussed.The initial value problems for the first-order NDDE with constant or proportional delay have been solved using the proposed block method.Some numerical results for the proposed method have been presented to prove the adaptability and applicability of the proposed method in solving NDDE.The proposed method is proved to be comparable with the other existing methods.It is assumed to be reliable and efficient for solving the first-order NDDE with constant or proportional delay.
文摘This paper focuses on the numerical stability of the block θ methods adapted to differential equations with a delay argument. For the block θ methods, an interpolation procedure is introduced which leads to the numerical processes that satisfy an important asymptotic stability condition related to the class of test problems y′(t)=ay(t)+by(t-τ) with a,b∈C, Re(a)<-|b| and τ>0. We prove that the block θ method is GP stable if and only if the method is A stable for ordinary differential equations. Furthermore, it is proved that the P and GP stability are equivalent for the block θ method.
基金The research of this author was supported by the National Natural Science Foundation of China,the JiangsuProvince Natural Science Foundation,the Jiangsu Province"333Engineering" Foundation and the Jiangsu Province"Qinglan Engineering" Foundation
文摘The symmetric linear system gives us many simplifications and a possibility to adapt the computations to the computer at hand in order to achieve better performance. The aim of this paper is to consider the block bidiagonalization methods derived from a symmetric augmented multiple linear systems and make a comparison with the block GMRES and block biconjugate gradient methods.
基金the National Natural Science Foundation of China(No.51478444,51574115 and 51774121)
文摘Based on the limit analysis upper bound method, a new mechanism of soil slope failure has been proposed which was consisted of plastic shear zone and rigid block zone. The different zones interface were regarded as discontinuity lines. Two sliding blocks of the slope were also incorporated horizontal seismic force and vertical gravity force. The velocities and forces were analyzed in two blocks, and the expression of velocity discontinuities was derived according to the principle of incompressibility. The external force done work for the blocks and the internal energy dissipated of the plastic shear zone and the velocity discontinuous were solved.The stability ratios were derived for the height of two-level slope with different rates to involve seismic and no seismic. The present stability ratios were compared to the previous study, which showed the superiority of the mechanism and the rationality of the analysis. The critical height of the slope can provide a theoretical basis for slope support and design.
基金The project supported by the National Natural Science Foundation of China
文摘Based on elementary group theory, the block pivot methods for solving two-dimensional elastic frictional contact problems are presented in this paper. It is proved that the algorithms converge within a finite number of steps when the friction coefficient is ''relative small''. Unlike most mathematical programming methods for contact problems, the block pivot methods permit multiple exchanges of basic and nonbasic variables.
文摘This paper presents a class of r-point (r+1)st-order A-stable one-block methods with damping at the infinite point (DIAOB r,r+1 ). Under the conditions of the same order, A-stability, operation count (at each iterative step) and storage space are the same as the methods in , the methods in the paper improve the stability in a neighborhood at the infinite point. And, by using the OOPI method , it possesses much faster rate of convergence for solving systems of nonlinear equations produced by the DIAOB r,r+1 .
文摘Iterative methods that take advantage of efficient block operations and block communications are popular research topics in parallel computation. These methods are especially important on Massively Parallel Processors (MPP). This paper presents a block variant of the GMRES method for solving general unsymmetric linear systems. It is shown that the new algorithm with block size s, denoted by BVGMRES(s,m), is theoretically equivalent to the GMRES(s. m) method. The numerical results show that this algorithm can be more efficient than the standard GMRES method on a cache based single CPU computer with optimized BLAS kernels. Furthermore, the gain in efficiency is more significant on MPPs due to both efficient block operations and efficient block data communications. Our numerical results also show that in comparison to the standard GMRES method, the more PEs that are used on an MPP, the more efficient the BVGMRES(s,m) algorithm is.
基金supported by the Innovation Platform Construction of Qinghai Province(No.2016-ZJ-Y04)the Basic Research Program of Qinghai Province(No.2016-ZJ-740)
文摘Text extraction is an important initial step in digitizing the historical documents. In this paper, we present a text extraction method for historical Tibetan document images based on block projections. The task of text extraction is considered as text area detection and location problem. The images are divided equally into blocks and the blocks are filtered by the information of the categories of connected components and corner point density. By analyzing the filtered blocks' projections, the approximate text areas can be located, and the text regions are extracted. Experiments on the dataset of historical Tibetan documents demonstrate the effectiveness of the proposed method.
文摘OBJECTIVE: To compare the clinical effect of brachial plexus block with "One Injection Two Points" guided under ultrasound and the conventional method guiding by ultrasound. METHODS: 70 patients were randomized evenly into 2 groups, with 35 patients in each group, while the Experiment Group(Group B) received One Injection Two Points" method, the Control Group(Group A) received the conventional method.The nerve block every 5 s, the success rate of anesthesia, the dosage of local anesthetics, second remedial anesthesia, adverse reactions, etc.were recorded. RESULTS: Group B was superior to group A in the success rate of anesthesia; There were 6 patients in group A who required constant pump injection of Remifentanil to remedy, while no patients in Group B needed remedy treatment. There were no serious adverse reactions in both groups.CONCLUSIONS: The clinical effect of brachial plexus block with "One Injection Two Points" method guided under ultrasoundguiding by ultrasound was superior to that of the conventional method.
