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New Numerical Integration Formulations for Ordinary Differential Equations
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作者 Serdar Beji 《Advances in Pure Mathematics》 2024年第8期650-666,共17页
An entirely new framework is established for developing various single- and multi-step formulations for the numerical integration of ordinary differential equations. Besides polynomials, unconventional base-functions ... An entirely new framework is established for developing various single- and multi-step formulations for the numerical integration of ordinary differential equations. Besides polynomials, unconventional base-functions with trigonometric and exponential terms satisfying different conditions are employed to generate a number of formulations. Performances of the new schemes are tested against well-known numerical integrators for selected test cases with quite satisfactory results. Convergence and stability issues of the new formulations are not addressed as the treatment of these aspects requires a separate work. The general approach introduced herein opens a wide vista for producing virtually unlimited number of formulations. 展开更多
关键词 Single- and Multi-Step Numerical Integration Unconventional Base-Functions ordinary differential equations
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On Bifurcations of an Ordinary Differential Equation
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作者 LI Chang-pin College of Sciences, Shanghai University, Shanghai 200436, China 《Advances in Manufacturing》 SCIE CAS 2000年第S1期4-6,共3页
Biforations of an ordinary differential equation with two-point boundary value condition are investigated. Using the singularity theory based on the Liapunov-Schmidt reduction, we have obtained some characterization r... Biforations of an ordinary differential equation with two-point boundary value condition are investigated. Using the singularity theory based on the Liapunov-Schmidt reduction, we have obtained some characterization results. 展开更多
关键词 BIFURCATIONS ordinary differential equation singularity theory Liapunov-Schlnidt reduction
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Interval of effective time-step size for the numerical computation of nonlinear ordinary differential equations
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作者 CAO Jing LI Jian-Ping ZHANG Xin-Yuan 《Atmospheric and Oceanic Science Letters》 CSCD 2017年第1期17-20,共4页
The computational uncertainty principle states that the numerical computation of nonlinear ordinary differential equations(ODEs) should use appropriately sized time steps to obtain reliable solutions.However,the int... The computational uncertainty principle states that the numerical computation of nonlinear ordinary differential equations(ODEs) should use appropriately sized time steps to obtain reliable solutions.However,the interval of effective step size(IES) has not been thoroughly explored theoretically.In this paper,by using a general estimation for the total error of the numerical solutions of ODEs,a method is proposed for determining an approximate IES by translating the functions for truncation and rounding errors.It also illustrates this process with an example.Moreover,the relationship between the IES and its approximation is found,and the relative error of the approximation with respect to the IES is given.In addition,variation in the IES with increasing integration time is studied,which can provide an explanation for the observed numerical results.The findings contribute to computational step-size choice for reliable numerical solutions. 展开更多
关键词 ordinary differential equations interval of effective step size computational uncertainty principle integration time relative error
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Existence and Multiplicities of Solutions for Asymptotically Linear Ordinary Differential Equations Satisfying Sturm-Liouville BVPs with Resonance
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作者 Keqiang Li Ronghua Tong 《Journal of Applied Mathematics and Physics》 2019年第5期1197-1211,共15页
In this paper, we prove existence and multiplicities of solutions for asymptotically linear ordinary differential equations satisfying Sturm-Liouville boundary value conditions with resonance. Adding assumption H3 tha... In this paper, we prove existence and multiplicities of solutions for asymptotically linear ordinary differential equations satisfying Sturm-Liouville boundary value conditions with resonance. Adding assumption H3 that is similar to (LL) in Theorem 1.1, by index theory and Morse theory, we obtain more nontrivial solutions. 展开更多
关键词 Asymptotically Linear ordinary differential equations with RESONANCE Multiple SOLUTIONS STURM-LIOUVILLE Boundary Value Problem Index theory Morse theory
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Stability Analysis of a Numerical Integrator for Solving First Order Ordinary Differential Equation
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作者 Samuel Olukayode Ayinde Adesoji Abraham Obayomi Funmilayo Sarah Adebayo 《Journal of Applied Mathematics and Physics》 2017年第11期2196-2204,共9页
In this paper, we used an interpolation function to derive a Numerical Integrator that can be used for solving first order Initial Value Problems in Ordinary Differential Equation. The numerical quality of the Integra... In this paper, we used an interpolation function to derive a Numerical Integrator that can be used for solving first order Initial Value Problems in Ordinary Differential Equation. The numerical quality of the Integrator has been analyzed to authenticate the reliability of the new method. The numerical test showed that the finite difference methods developed possess the same monotonic properties with the analytic solution of the sampled Initial Value Problems. 展开更多
关键词 NUMERICAL INTEGRATOR Autonomous and NON-AUTONOMOUS ordinary differential equation INITIAL Value Problems Stability Analysis
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INTEGRABLE TYPES OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION SETS OF HIGHER ORDERS
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作者 汤光宋 原存德 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1995年第9期883-890,共8页
Because of the extensive applications of nonlinear ordinary differential equation in physics,mechanics and cybernetics,there have been many papers on the exact solution to differential equation in some major publicati... Because of the extensive applications of nonlinear ordinary differential equation in physics,mechanics and cybernetics,there have been many papers on the exact solution to differential equation in some major publications both at home and abroad in recent years Based on these papers and in virtue of Leibniz formula,and transformation set technique,this paper puts forth the solution to nonlinear ordinary differential equation set of higher-orders, moveover,its integrability is proven.The results obtained are the generalization of those in the references. 展开更多
关键词 nonlinear ordinary differential equation set.transformation set.integrable type
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Conformal invariance and integration of first-order differential equations 被引量:7
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作者 何光 梅凤翔 《Chinese Physics B》 SCIE EI CAS CSCD 2008年第8期2764-2766,共3页
This paper studies a conformal invariance and an integration of first-order differential equations. It obtains the corresponding infinitesimal generators of conformal invariance by using the symmetry of the differenti... This paper studies a conformal invariance and an integration of first-order differential equations. It obtains the corresponding infinitesimal generators of conformal invariance by using the symmetry of the differential equations, and expresses the differential equations by the equations of a Birkhoff system or a generalized Birkhoff system. If the infinitesimal generators are those of a Noether symmetry, the conserved quantity can be obtained by using the Noether theory of the Birkhoff system or the generalized Birkhoff system. 展开更多
关键词 differential equation conformal invariance Noether theory INTEGRATION
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A Birkhoff-Noether method of solving differential equations 被引量:2
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作者 尚玫 郭永新 梅凤翔 《Chinese Physics B》 SCIE EI CAS CSCD 2007年第2期292-295,共4页
In this paper, a Birkhoff-Noether method of solving ordinary differential equations is presented. The differential equations can be expressed in terms of Birkhoff's equations. The first integrals for differential equ... In this paper, a Birkhoff-Noether method of solving ordinary differential equations is presented. The differential equations can be expressed in terms of Birkhoff's equations. The first integrals for differential equations can be found by using the Noether theory for Birkhoffian systems. Two examples are given to illustrate the application of the method. 展开更多
关键词 differential equation Birkhoffian system Noether theory first integral
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On Trigonometric Numerical Integrator for Solving First Order Ordinary Differential Equation 被引量:1
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作者 A. A. Obayomi S. O. Ayinde O. M. Ogunmiloro 《Journal of Applied Mathematics and Physics》 2019年第11期2564-2578,共15页
In this paper, we used an interpolation function with strong trigonometric components to derive a numerical integrator that can be used for solving first order initial value problems in ordinary differential equation.... In this paper, we used an interpolation function with strong trigonometric components to derive a numerical integrator that can be used for solving first order initial value problems in ordinary differential equation. This numerical integrator has been tested for desirable qualities like stability, convergence and consistency. The discrete models have been used for a numerical experiment which makes us conclude that the schemes are suitable for the solution of first order ordinary differential equation. 展开更多
关键词 NUMERICAL INTEGRATOR ordinary differential equation INITIAL Value Problems Stability Analysis NONSTANDARD METHODS INTERPOLATION METHODS
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INSTABILITY OF SOLUTION FOR A CLASS OF THE THIRD ORDER NONLINEAR DIFFERENTIAL EQUATION 被引量:1
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作者 卢德渊 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1995年第12期1185-1200,共16页
In Ref [1] the asymptotic stability of nonlinear slowly changing system has been discussed .In Ref [2] the instability of solution for the order linear differential equaiton with varied coefficient has been discussed ... In Ref [1] the asymptotic stability of nonlinear slowly changing system has been discussed .In Ref [2] the instability of solution for the order linear differential equaiton with varied coefficient has been discussed .In this paper,we have discussed instability of solution for a class of the third order nonlinear diffeential equation by means of the metod of Refs [1] and [2] . 展开更多
关键词 ordinary difterential cquation motive stability theory .nonlinear differential equation
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INSTABILITY OF SOLUTION FOR THE FOURTH ORDER LINEAR DIFFERENTIAL EQUATION WITH VARIED COEFFICIENT
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作者 卢德渊 廖宗璜 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1993年第5期481-497,共17页
In this paper, we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part,... In this paper, we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part, by means of Liapunov's second method. 展开更多
关键词 ordinary differential equation motive stability theory linear differential equation with varied coefficient
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Lagrange-Noether method for solving second-order differential equations 被引量:1
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作者 吴惠彬 吴润衡 《Chinese Physics B》 SCIE EI CAS CSCD 2009年第9期3647-3650,共4页
The purpose of this paper is to provide a new method called the Lagrange-Noether method for solving second-order differential equations. The method is, firstly, to write the second-order differential equations complet... The purpose of this paper is to provide a new method called the Lagrange-Noether method for solving second-order differential equations. The method is, firstly, to write the second-order differential equations completely or partially in the form of Lagrange equations, and secondly, to obtain the integrals of the equations by using the Noether theory of the Lagrange system. An example is given to illustrate the application of the result. 展开更多
关键词 differential equation Lagrange equation Noether theory INTEGRAL
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Noether's and Poisson's methods for solving differential equation x_s^((m))=F_s(t,x_k^((m-2)) ,x_k^((m-1)))
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作者 何光 梅凤翔 《Chinese Physics B》 SCIE EI CAS CSCD 2008年第3期822-824,共3页
This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether me... This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether method and the Poisson method. Then the solution of the higher-order equation can be obtained by integrating the solution of the second-order equation. 展开更多
关键词 Noether's method Poisson's method higher order ordinary differential equation integration
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精细积分方法的发展与扩展应用 被引量:1
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作者 姚伟岸 高强 +1 位作者 谭述君 吴锋 《计算力学学报》 CAS CSCD 北大核心 2024年第1期2-25,共24页
钟万勰院士于1991年首先提出计算矩阵指数的精细积分方法,其要点是2N类算法和增量存储。精细积分方法可给出矩阵指数在计算机意义上的精确解,为常微分方程的数值计算提供了高精度、高稳定性的算法,现已成功应用于结构动力响应、随机振... 钟万勰院士于1991年首先提出计算矩阵指数的精细积分方法,其要点是2N类算法和增量存储。精细积分方法可给出矩阵指数在计算机意义上的精确解,为常微分方程的数值计算提供了高精度、高稳定性的算法,现已成功应用于结构动力响应、随机振动、热传导以及最优控制等众多领域。本文首先介绍矩阵指数精细积分方法的提出、基本思想和发展;然后依次介绍在时不变/时变线性微分方程、非线性微分方程以及大规模问题求解中发展起来的各种精细积分方法,分析了其优缺点和适用范围;最后介绍了精细积分方法的基本思想在两点边值问题、椭圆函数和病态代数方程等问题的扩展应用,进一步展示了该思想的特色。 展开更多
关键词 精细积分方法 矩阵指数 常微分方程 时程积分
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NEW PROGRESS IN THEORY OF SURFACES DEFINED BY ORDINARY DIFFERENTIAL EQUATIONS~*
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作者 秦元勋 《Chinese Science Bulletin》 SCIE EI CAS 1989年第10期800-803,共4页
Here singular points of the system (E_n~*) includes both finite and infinite singular points defined by H. Poincare, but extended to complex domain.
关键词 nonlinear ordinary differential equationS strong rooted THEOREM general INTEGRAL representation THEOREM HILBERT number N(n).
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THEORY OF SINGULAR POINTS OF ORDINARY DIFFERENTIAL EQUATIONS IN COMPLEX DOMAIN
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作者 秦元勋 赵怀忠 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 1992年第4期298-317,共20页
In this paper, the topological of integral surfaces near certain of Lyapunov type singularpoints and certain type of nodes of ordinary differential equations in complex domain are studied.We introduce Briot-Bouquet tr... In this paper, the topological of integral surfaces near certain of Lyapunov type singularpoints and certain type of nodes of ordinary differential equations in complex domain are studied.We introduce Briot-Bouquet transformation, in order to study the topological structure of integralsurfaces near higher order singular points. At last we give an estimate of the makimum numberof isolated limit integral surfaces passing through certain type of higher order singular points. 展开更多
关键词 theory of SINGULAR POINTS of ordinary differential equationS IN COMPLEX DOMAIN BZB 切二
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一类Hilfer分数阶微分方程的平方可积解
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作者 周乐 郝晓玲 《内蒙古大学学报(自然科学版)》 CAS 2024年第3期225-231,共7页
对一类Hilfer型分数阶Sturm-Liouville方程,证明了其Weyl-Titchmarsh理论,并得出了定义在奇异区间上的此类方程的解的平方可积性。
关键词 分数阶微分方程 平方可积解 Weyl-Titchmarsh理论 Hilfer导数
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分数阶共振向量边值问题解的存在性
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作者 司换敏 江卫华 王璐瑶 《河北师范大学学报(自然科学版)》 CAS 2024年第4期345-354,共10页
研究了一类带有系数矩阵的分数阶共振边值问题.首先,定义2个合适的Banach空间;然后,在Banach空间中构造恰当的算子并使用仿伪逆矩阵的性质及Mawhin的重合度理论,得到了其解的存在性;最后,给出一个例子来验证主要结果.
关键词 常微分方程 向量边值问题 仿伪逆矩阵 Mawhin的重合度理论 共振
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一阶常微分方程的知识图谱
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作者 黎平 杨晓珍 《凯里学院学报》 2024年第3期1-9,共9页
一阶常微分方程的方程类型较多,解法不一.在学习的过程中会因为方程类型的判断不准确,导致学习困难.本文通过构造一阶常微分方程的知识图谱,将繁杂的知识进行关联与梳理,呈现常见一阶常微分方程的方程类型和解法之间的相互关系,并以实... 一阶常微分方程的方程类型较多,解法不一.在学习的过程中会因为方程类型的判断不准确,导致学习困难.本文通过构造一阶常微分方程的知识图谱,将繁杂的知识进行关联与梳理,呈现常见一阶常微分方程的方程类型和解法之间的相互关系,并以实例进行说明.构建微分方程的知识图谱可以对微分方程中的概念、方法和技巧等进行全面和系统地整合,呈现出方程之间的内在联系和逻辑关系,可以培养学生的逻辑思维能力,帮助学生更加深入和全面地学习微分方程的知识,提高学习效果. 展开更多
关键词 知识图谱 常微分方程 常数变易法 积分因子
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关于二阶Fermat型常微分方程的整函数解
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作者 张宇 杨刘 《数学杂志》 2024年第4期317-330,共14页
本文研究了二阶Fermat型常微分方程(a_(1)f+b_(1)f′+c_(1)f")^(2)+(a_(2)f+b_(2)f′+c_(2)f")^(2)=γ的整函数解的问题,其中γ是C上的整函数.利用Nevanlinna值分布理论的方法,获得了方程存在整函数解的充要条件,并且给出了... 本文研究了二阶Fermat型常微分方程(a_(1)f+b_(1)f′+c_(1)f")^(2)+(a_(2)f+b_(2)f′+c_(2)f")^(2)=γ的整函数解的问题,其中γ是C上的整函数.利用Nevanlinna值分布理论的方法,获得了方程存在整函数解的充要条件,并且给出了解的表达形式. 展开更多
关键词 二阶Fermat型复微分方程 NEVANLINNA值分布理论 整函数
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