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A UNIFORMLY CONVERGENT SECOND ORDER DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM
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作者 郭雯 林鹏程 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1989年第3期231-241,共11页
In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniform... In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme. 展开更多
关键词 exp A UNIFORMLY convergent SECOND order DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM
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Self-adaptive strategy for one-dimensional finite element method based on EEP method with optimal super-convergence order 被引量:4
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作者 袁驷 邢沁妍 +1 位作者 王旭 叶康生 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2008年第5期591-602,共12页
Based on the newly-developed element energy projection (EEP) method with optimal super-convergence order for computation of super-convergent results, an improved self-adaptive strategy for one-dimensional finite ele... Based on the newly-developed element energy projection (EEP) method with optimal super-convergence order for computation of super-convergent results, an improved self-adaptive strategy for one-dimensional finite element method (FEM) is proposed. In the strategy, a posteriori errors are estimated by comparing FEM solutions to EEP super-convergent solutions with optimal order of super-convergence, meshes are refined by using the error-averaging method. Quasi-FEM solutions are used to replace the true FEM solutions in the adaptive process. This strategy has been found to be simple, clear, efficient and reliable. For most problems, only one adaptive step is needed to produce the required FEM solutions which pointwise satisfy the user specified error tolerances in the max-norm. Taking the elliptical ordinary differential equation of the second order as the model problem, this paper describes the fundamental idea, implementation strategy and computational algorithm and representative numerical examples are given to show the effectiveness and reliability of the proposed approach. 展开更多
关键词 finite element method (FEM) self-adaptive solution super-convergence optimal convergence order element energy projection condensed shape functions
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A Family of Fifth-order Iterative Methods for Solving Nonlinear Equations 被引量:4
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作者 Liu Tian-Bao Cai Hua Li Yong 《Communications in Mathematical Research》 CSCD 2013年第3期255-260,共6页
In this paper, we present and analyze a family of fifth-order iterative methods free from second derivative for solving nonlinear equations. It is established that the family of iterative methods has convergence order... In this paper, we present and analyze a family of fifth-order iterative methods free from second derivative for solving nonlinear equations. It is established that the family of iterative methods has convergence order five. Numerical examples show that the new methods are comparable with the well known existing methods and give better results in many aspects. 展开更多
关键词 Newton's method iterative method nonlinear equation order of convergence
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Dynamical Comparison of Several Third-Order Iterative Methods for Nonlinear Equations 被引量:2
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作者 Obadah Said Solaiman Samsul Ariffin Abdul Karim Ishak Hashim 《Computers, Materials & Continua》 SCIE EI 2021年第5期1951-1962,共12页
There are several ways that can be used to classify or compare iterative methods for nonlinear equations,for instance;order of convergence,informational efficiency,and efficiency index.In this work,we use another way,... There are several ways that can be used to classify or compare iterative methods for nonlinear equations,for instance;order of convergence,informational efficiency,and efficiency index.In this work,we use another way,namely the basins of attraction of the method.The purpose of this study is to compare several iterative schemes for nonlinear equations.All the selected schemes are of the third-order of convergence and most of them have the same efficiency index.The comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different degrees.As a comparison,we determine the CPU time(in seconds)needed by each scheme to obtain the basins of attraction,besides,we illustrate the area of convergence of these schemes by finding the number of convergent and divergent points in a selected range for all methods.Comparisons confirm the fact that basins of attraction differ for iterative methods of different orders,furthermore,they vary for iterative methods of the same order even if they have the same efficiency index.Consequently,this leads to the need for a new index that reflects the real efficiency of the iterative scheme instead of the commonly used efficiency index. 展开更多
关键词 Nonlinear equations iterative methods basins of attraction order of convergence
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An Iterative Scheme of Arbitrary Odd Order and Its Basins of Attraction for Nonlinear Systems 被引量:2
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作者 Obadah Said Solaiman Ishak Hashim 《Computers, Materials & Continua》 SCIE EI 2021年第2期1427-1444,共18页
In this paper,we propose a fifth-order scheme for solving systems of nonlinear equations.The convergence analysis of the proposed technique is discussed.The proposed method is generalized and extended to be of any odd... In this paper,we propose a fifth-order scheme for solving systems of nonlinear equations.The convergence analysis of the proposed technique is discussed.The proposed method is generalized and extended to be of any odd order of the form 2n1.The scheme is composed of three steps,of which the first two steps are based on the two-step Homeier’s method with cubic convergence,and the last is a Newton step with an appropriate approximation for the derivative.Every iteration of the presented method requires the evaluation of two functions,two Fréchet derivatives,and three matrix inversions.A comparison between the efficiency index and the computational efficiency index of the presented scheme with existing methods is performed.The basins of attraction of the proposed scheme illustrated and compared to other schemes of the same order.Different test problems including large systems of equations are considered to compare the performance of the proposed method according to other methods of the same order.As an application,we apply the new scheme to some real-life problems,including the mixed Hammerstein integral equation and Burgers’equation.Comparisons and examples show that the presented method is efficient and comparable to the existing techniques of the same order. 展开更多
关键词 System of nonlinear equations root finding method iterative method order of convergence Burgers’equation
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ERRATA TO: “λ-Statistical Convergence of Order α” Acta Mathematica Scientia 2011, 31B(3): 953-959 被引量:1
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作者 R. olak . A. Bektas 《Acta Mathematica Scientia》 SCIE CSCD 2011年第5期2099-2100,共2页
The sequences defined in Example 3 and Example 4 do not serve our purpose for any λ = (λn). Because this sequences are just the sequences x = (xk) = (k) and x = (xk) = (1) respectively and any term of thes... The sequences defined in Example 3 and Example 4 do not serve our purpose for any λ = (λn). Because this sequences are just the sequences x = (xk) = (k) and x = (xk) = (1) respectively and any term of these sequences can not be 0. In this short not we give Example 3* and Example 4* to show that the inclusions given in Theorem 2.4 and Theorem 2.9 are strict for some λ = (λn) , α and β such that 0 α β ≤ 1. 展开更多
关键词 Statistical Convergence of order Acta Mathematica Scientia 2011 ERRATA TO
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On the Location of Zeros of Higher Order Differential Equation 被引量:2
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作者 CHEN Yu-xian WU Zhao-jun 《Chinese Quarterly Journal of Mathematics》 CSCD 2010年第1期92-97,共6页
In this paper, by using the Nevanlinna Theory on angular domain, we establish a theorem which concerns the growth of entire function and his zero. As an application, we survey the location of zero of higher order diff... In this paper, by using the Nevanlinna Theory on angular domain, we establish a theorem which concerns the growth of entire function and his zero. As an application, we survey the location of zero of higher order differential equation, which can be regarded as an alternating but precise version of Wu and Yi. 展开更多
关键词 second order exponent convergence Nevanlinna theory higher order differential equation
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On the best convergence order of a new class of triangular summation operators
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作者 孟佳娜 赵连霞 《Journal of Shanghai University(English Edition)》 CAS 2006年第5期399-401,共3页
In this paper, a new class of triangular summation operators based on the equidistant nodes was constructed. It is proved that this class of operators converges uniformly to arbitrary continuous fimctions with the per... In this paper, a new class of triangular summation operators based on the equidistant nodes was constructed. It is proved that this class of operators converges uniformly to arbitrary continuous fimctions with the period 2π on the whole axis, Fttrthermore, the best approximation order and the highest convergence order are obtained. In contrast to certain operators constructed by Bernstein and Kis in the previous works, the convergence properties of the new operator constructed in this paper are superior. 展开更多
关键词 triangular summation operator uniform convergence the best approxdmation order the highest convergence order
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Efficient Numerical Scheme for Solving Large System of Nonlinear Equations
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作者 Mudassir Shams Nasreen Kausar +2 位作者 Shams Forruque Ahmed Irfan Anjum Badruddin Syed Javed 《Computers, Materials & Continua》 SCIE EI 2023年第3期5331-5347,共17页
A fifth-order family of an iterative method for solving systems of nonlinear equations and highly nonlinear boundary value problems has been developed in this paper.Convergence analysis demonstrates that the local ord... A fifth-order family of an iterative method for solving systems of nonlinear equations and highly nonlinear boundary value problems has been developed in this paper.Convergence analysis demonstrates that the local order of convergence of the numerical method is five.The computer algebra system CAS-Maple,Mathematica,or MATLAB was the primary tool for dealing with difficult problems since it allows for the handling and manipulation of complex mathematical equations and other mathematical objects.Several numerical examples are provided to demonstrate the properties of the proposed rapidly convergent algorithms.A dynamic evaluation of the presented methods is also presented utilizing basins of attraction to analyze their convergence behavior.Aside from visualizing iterative processes,this methodology provides useful information on iterations,such as the number of diverging-converging points and the average number of iterations as a function of initial points.Solving numerous highly nonlinear boundary value problems and large nonlinear systems of equations of higher dimensions demonstrate the performance,efficiency,precision,and applicability of a newly presented technique. 展开更多
关键词 Nonlinear equations convergence order boundary value problem computational time basins of attraction converging points
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Two Families of Multipoint Root-Solvers Using Inverse Interpolation with Memory
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作者 Zhongli Liu Quan Zheng 《Journal of Applied Mathematics and Physics》 2023年第3期746-759,共14页
In this paper, a general family of derivative-free n + 1-point iterative methods using n + 1 evaluations of the function and a general family of n-point iterative methods using n evaluations of the function and only o... In this paper, a general family of derivative-free n + 1-point iterative methods using n + 1 evaluations of the function and a general family of n-point iterative methods using n evaluations of the function and only one evaluation of its derivative are constructed by the inverse interpolation with the memory on the previous step for solving the simple root of a nonlinear equation. The order and order of convergence of them are proved respectively. Finally, the proposed methods and the basins of attraction are demonstrated by the numerical examples. 展开更多
关键词 Nonlinear Equation General Multipoint Iteration Inverse Interpolation order of Convergence Basin of Attraction
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Ostrowski’s Method for Solving Nonlinear Equations and Systems
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作者 Christian Beleña Postigo 《Journal of Mechanics Engineering and Automation》 2023年第1期1-6,共6页
The dynamic characteristics and the efficiency of the Ostrowski’s method allow it to be crowned as an excellent tool for solving nonlinear problems.This article shows different versions of the classic method that all... The dynamic characteristics and the efficiency of the Ostrowski’s method allow it to be crowned as an excellent tool for solving nonlinear problems.This article shows different versions of the classic method that allow it to be applied to a wide range of engineering problems.Among them stands out the derivative-free definition applying divided differences,the introduction of memory and its extension to the resolution of nonlinear systems of equations.All of these versions are compared in a numerical simulations section where the results obtained are compared with other classic methods. 展开更多
关键词 Iterative methods nonlinear equations convergence order stability.
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On Triangle Interpolation Approximation of the Double Function 被引量:3
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作者 袁学刚 韩友发 《Chinese Quarterly Journal of Mathematics》 CSCD 2000年第4期61-65,共5页
本文以两组不同的节点构造了一个组合型的二元三角插值多项式算子Lmn(f;x,y),并且研究了二元连续周期函数对这个算子的收敛性及收敛阶的估计等问题。
关键词 double triangle interpolation polymial convergence order converge uniforml?
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Error Analysis on Corrector Formula for Rectangular Rule 被引量:1
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作者 XIAO Ze-chang DU Yue-peng 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2008年第2期270-275,共6页
This paper presents truncation errors among Corrector Formula for left Rectangular rule and Corrector Formula for middle Rectangular rule respectively. It also displays an analysis on convergence order of compound cor... This paper presents truncation errors among Corrector Formula for left Rectangular rule and Corrector Formula for middle Rectangular rule respectively. It also displays an analysis on convergence order of compound corrector formulas for rectangular rule. Examples of numerical calculation have validated theoretical analysis. 展开更多
关键词 numerical integration algebraic accuracy corrector formula truncation error convergence order
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Improved Cotes Formula and Error Analysis 被引量:1
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作者 DU Yue-peng XIAO Ze-chang 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2008年第3期458-461,共4页
The truncation error of improved Cotes formula is presented in this paper. It also displays an analysis on convergence order of improved Cotes formula. Examples of numerical calculation is given in the end.
关键词 numerical integration algebraic accuracy truncation error convergence order
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Approximation to Continuous Functions by a Kind of Interpolation Polynomials 被引量:2
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作者 袁学刚 王德辉 《Northeastern Mathematical Journal》 CSCD 2001年第1期39-44,共6页
In this paper, an interpolation polynomial operator F n(f; l,x) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈C b [-1,1] ... In this paper, an interpolation polynomial operator F n(f; l,x) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function f(x)∈C b [-1,1] (0≤b≤l) F n(f; l,x) converges to f(x) uniformly, where l is an odd number. 展开更多
关键词 interpolation polynomial uniform convergence approximation order the highest convergence order
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Two-stage Milstein Methods for Stochastic Differential Equations 被引量:1
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作者 王鹏 吕显瑞 柳振鑫 《Northeastern Mathematical Journal》 CSCD 2008年第1期63-76,共14页
In this paper we discuss two-stage Miistein methods for solving Ito stochastic differential equations (SDEs). Six fully explicit methods (TSM 1 -- TSM 6) are given in this paper. Their order of strong convergence ... In this paper we discuss two-stage Miistein methods for solving Ito stochastic differential equations (SDEs). Six fully explicit methods (TSM 1 -- TSM 6) are given in this paper. Their order of strong convergence is proved. The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of Ito SDEs. 展开更多
关键词 stochastic differential equation Euler-Maruyama method Milstein method STABILITY strong convergence order
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Note on a New Construction of Kantorovich Form q-Bernstein Operators Related to Shape Parameter λ 被引量:1
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作者 Qingbo Cai Resat Aslan 《Computer Modeling in Engineering & Sciences》 SCIE EI 2022年第3期1479-1493,共15页
The main purpose of this paper is to introduce some approximation properties of a Kantorovich kind q-Bernstein operators related to B′ezier basis functions with shape parameterλ∈[−1,1].Firstly,we compute some basic... The main purpose of this paper is to introduce some approximation properties of a Kantorovich kind q-Bernstein operators related to B′ezier basis functions with shape parameterλ∈[−1,1].Firstly,we compute some basic results such as moments and central moments,and derive the Korovkin type approximation theorem for these operators.Next,we estimate the order of convergence in terms of the usual modulus of continuity,for the functions belong to Lipschitz-type class and Peetre’s K-functional,respectively.Lastly,with the aid of Maple software,we present the comparison of the convergence of these newly defined operators to the certain function with some graphical illustrations and error estimation table. 展开更多
关键词 Q-CALCULUS q)-Bernstein polynomials order of convergence Lipschitz-type function Peetre’s K-functional
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On Double Revised Nodes of S N Bernstein Interpolation Process of the Third Type
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作者 CHANG Yu-bao WEI Ping YUAN Xue-gang 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2008年第4期559-564,共6页
In this work, the well-known problem put forward by S N Bernstein in 1930 is studied in a deep step. An operator is constructed by revising double interpolation nodes. It is proved that the operator converges to arbit... In this work, the well-known problem put forward by S N Bernstein in 1930 is studied in a deep step. An operator is constructed by revising double interpolation nodes. It is proved that the operator converges to arbitrary continuous functions uniformly and the convergence order is the best. 展开更多
关键词 interpolation polynomial uniform convergence the highest convergence order S N Bernstein problem
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Computer Oriented Numerical Scheme for Solving Engineering Problems
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作者 Mudassir Shams Naila Rafiq +2 位作者 Nasreen Kausar Nazir Ahmad Mir Ahmad Alalyani 《Computer Systems Science & Engineering》 SCIE EI 2022年第8期689-701,共13页
In this study,we construct a family of single root finding method of optimal order four and then generalize this family for estimating of all roots of non-linear equation simultaneously.Convergence analysis proves tha... In this study,we construct a family of single root finding method of optimal order four and then generalize this family for estimating of all roots of non-linear equation simultaneously.Convergence analysis proves that the local order of convergence is four in case of single root finding iterative method and six for simultaneous determination of all roots of non-linear equation.Some non-linear equations are taken from physics,chemistry and engineering to present the performance and efficiency of the newly constructed method.Some real world applications are taken from fluid mechanics,i.e.,fluid permeability in biogels and biomedical engineering which includes blood Rheology-Model which as an intermediate result give some nonlinear equations.These non-linear equations are then solved using newly developed simultaneous iterative schemes.Newly developed simultaneous iterative schemes reach to exact values on initial guessed values within given tolerance,using very less number of function evaluations in each step.Local convergence order of single root finding method is computed using CAS-Maple.Local computational order of convergence,CPU-time,absolute residuals errors are calculated to elaborate the efficiency,robustness and authentication of the iterative simultaneous method in its domain. 展开更多
关键词 Biomedical engineering convergence order iterative method CPU-time simultaneous method
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A New Third S.N.Bernstein Interpolation Polynomial
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作者 何甲兴 李笑牛 《Chinese Quarterly Journal of Mathematics》 CSCD 1998年第4期10-16, ,共7页
In this paper,a new third type S.N.Bernstein interpolation polynomial H n(f;x,r) with zeros of the Chebyshev ploynomial of the second kind is constructed. H n(f;x,r) converge uniformly on [-1,1] for any continuous fun... In this paper,a new third type S.N.Bernstein interpolation polynomial H n(f;x,r) with zeros of the Chebyshev ploynomial of the second kind is constructed. H n(f;x,r) converge uniformly on [-1,1] for any continuous function f(x) . The convergence order is the best order if \{f(x)∈C j[-1,1],\}0jr, where r is an odd natural number. 展开更多
关键词 uniform approximation the best convergence order Lagrange interpolation polynomial
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