THE NUMERICAL STABILITIES OF MULTIDERIVATIVE BLOCK METHOD

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)

High Order Block Method for Third Order ODEs

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.

Runge-Kutta Method and Bolck by Block Method to Solve Nonlinear Fredholm-Volterra Integral Equation with Continuous Kernel

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.

Two-Step Hybrid Block Method for Solving Nonlinear Jerk Equations

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.

Half-Step Continuous Block Method for the Solutions of Modeled Problems of Ordinary Differential Equations

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.

A New One-Twelfth Step Continuous Block Method for the Solution of Modeled Problems of Ordinary Differential Equations

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.

Numerical Solution for Neutral Delay Differential Equation of Constant or Proportional Type using Hybrid Block Method

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.

A Generalized Block-by-block Method for the System of Linear Volterra Integral Equations of the Second Kind

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.

THE NUMERICAL STABILITY OF THE BLOCK θ-METHODS FOR DELAY DIFFERENTIAL EQUATIONS

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.

BLOCK BIDIAGONALIZATION METHODS FOR MULTIPLE NONSYMMETRIC LINEAR SYSTEMS

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.

BLOCK PIVOT METHODS FOR SOLVING FRICTIONAL CONTACT PROBLEMS

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.

A CLASS OF r-POINT (r+1)ST-ORDER A-STABLE ONE-BLOCK METHODS WITH DAMPING AT THE INFINITE POINT

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 .

OPTIMIZATION AND METHOD OF DRAWING CONTROL IN BLOCK CAVING AT TONGKUANGYU MINE

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A BLOCK VARIANT OF THE GMRES METHOD FOR UNSYMMETRIC LINEAR SYSTEMS

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.

Clinical Observation on Brachial Plexus Block with “One Injection Two Points” Method Guiding by Ultrasound

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.

脉冲微分方程的一个修正block-by-block数值格式

本文利用修正的block-by-block方法针对脉冲微分方程构造了高阶数值格式.修正的block-by-block方法是传统的block-by-block方法的改进,其优点是除第一块外其余每块都能够解耦求解积分方程的高阶数值方法.首先,把脉冲微分方程等价转化为脉冲型积分方程,并利用修正的block-by-block方法进行离散,得到在两个相邻脉冲点中除第一块外其余每块都解耦的高阶数值格式.其次,利用离散的Grownwall不等式证明了数值解逼近精度为四阶.最后,一系列的数值算例验证了理论分析的正确性.

求解瞬态相干作用的Maxwell Block方程的一种矩阵方法

在一定的条件下我们可用矩阵的方法求解ＭａｘｗｅｌｌＢｌｏｃｋ 方程，这种矩阵的方法具有物理思路清晰、方法新颖的特点，并且所得到的结果与传统方法完全一致。

极坐标测量方法在Twin-Block矫治下颌后缩疗效分析中的应用

目的探讨极坐标测量分析方法用于Twin-Block矫治器矫治骨性Ⅱ类错疗效评价测量分析的临床应用价值。方法随机选取处于发育高峰期的骨性下颌后缩患者20例,男女各10例,采用Twin-Block矫治器矫治。对患者治疗前后的头颅侧位片采用极坐标测量分析方法,以S点为极坐标原点,前颅底平面(S-N)作极轴,建立极坐标测量体系,选取7个颌骨标志点A、Ptm、B、Po、Gn、Go、Ar,分别测量各标志点治疗前后极径和极角的大小,并作相应统计分析。结果 Ptm的极角和极径无明显改变,A点的极角和极径则均明显增大,说明Twin-Block矫治器对上颌骨生长无明显抑制作用;下颌标志点B、Po、Gn、Go点的极径在治疗后明显增大,但反映下颌骨改建方向的极角在治疗后则均无明显改变,说明Twin-Block矫治器对下颌骨的生长起到明显的促进作用,而对下颌骨生长方向没有明显影响。结论极坐标测量分析方法可以对颌骨标记点的生长量和生长方向作出准确的定量测量和定性分析,可弥补传统测量方法的不足;极坐标测量分析也表明,Twin-Block矫治器主要通过促进下颌骨的生长,而非通过抑制上颌骨的生长来调整上下颌骨矢状不调的位置关系,改善下颌后缩患者的侧貌美观。

一类分数阶Langevin方程block⁃by⁃block算法的数值分析

分数阶Langevin方程有重要的科学意义和工程应用价值,基于经典block⁃by⁃block算法,求解了一类含有Caputo导数的分数阶Langevin方程的数值解.Block⁃by⁃block算法通过引入二次Lagrange基函数插值,构造出逐块收敛的非线性方程组,通过在每一块耦合求得分数阶Langevin方程的数值解.在0<α<1条件下,应用随机Taylor展开证明block⁃by⁃block算法是3+α阶收敛的,数值试验表明在不同α和时间步长h取值下,block⁃by⁃block算法具有稳定性和收敛性,克服了现有方法求解分数阶Langevin方程速度慢精度低的缺点,表明block⁃by⁃block算法求解分数阶Langevin方程是高效的.