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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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A class of twostep continuity Runge-Kutta methods for solving singular delay differential equations and its convergence 被引量:1
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作者 Leng Xin Liu Degui +1 位作者 Song Xiaoqiu Chen Lirong 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2005年第4期908-916,共9页
An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods (TSCRK) is presented. Their construction, their order conditio... An idea of relaxing the effect of delay when computing the Runge-Kutta stages in the current step and a class of two-step continuity Runge-Kutta methods (TSCRK) is presented. Their construction, their order conditions and their convergence are studied. The two-step continuity Runge-Kutta methods possess good numerical stability properties and higher stage-order, and keep the explicit process of computing the Runge-Kutta stages. The numerical experiments show that the TSCRK methods are efficient. 展开更多
关键词 CONVERGENCE singular delay differential equations two-step continuity Runge-Kutta methods.
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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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An Adaptive Time-Step Backward Differentiation Algorithm to Solve Stiff Ordinary Differential Equations: Application to Solve Activated Sludge Models 被引量:2
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作者 Jamal Alikhani Bahareh Shoghli +1 位作者 Ujjal Kumar Bhowmik Arash Massoudieh 《American Journal of Computational Mathematics》 2016年第4期298-312,共15页
A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency ... A backward differentiation formula (BDF) has been shown to be an effective way to solve a system of ordinary differential equations (ODEs) that have some degree of stiffness. However, sometimes, due to high-frequency variations in the external time series of boundary conditions, a small time-step is required to solve the ODE system throughout the entire simulation period, which can lead to a high computational cost, slower response, and need for more memory resources. One possible strategy to overcome this problem is to dynamically adjust the time-step with respect to the system’s stiffness. Therefore, small time-steps can be applied when needed, and larger time-steps can be used when allowable. This paper presents a new algorithm for adjusting the dynamic time-step based on a BDF discretization method. The parameters used to dynamically adjust the size of the time-step can be optimally specified to result in a minimum computation time and reasonable accuracy for a particular case of ODEs. The proposed algorithm was applied to solve the system of ODEs obtained from an activated sludge model (ASM) for biological wastewater treatment processes. The algorithm was tested for various solver parameters, and the optimum set of three adjustable parameters that represented minimum computation time was identified. In addition, the accuracy of the algorithm was evaluated for various sets of solver parameters. 展开更多
关键词 Adaptive Time-step Backward differentiation Formula Activated Sludge Model Ordinary differential equation Stiffness Computation Time
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Solution of Nonlinear Integro Differential Equations by Two-Step Adomian Decomposition Method (TSAM)
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作者 Maryam Al-Mazmumy Safa O. Almuhalbedi 《International Journal of Modern Nonlinear Theory and Application》 2016年第4期248-255,共8页
The Adomian decomposition method (ADM) can be used to solve a wide range of problems and usually gets the solution in a series form. In this paper, we propose two-step Adomian Decomposition Method (TSAM) for nonlinear... The Adomian decomposition method (ADM) can be used to solve a wide range of problems and usually gets the solution in a series form. In this paper, we propose two-step Adomian Decomposition Method (TSAM) for nonlinear integro-differential equations that will facilitate the calculations. In this modification, compared to the standard Adomian decomposition method, the size of calculations was reduced. This modification also avoids computing Adomian polynomials. Numerical results are given to show the efficiency and performance of this method. 展开更多
关键词 Adomian Decomposition Method Nonlinear Volterraintegro-differential equations Nonlinear Fredholmintegro-differential equations TWO-step
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Suppressive Influence of Time- Space White Noise on the Explosion of Solutions of Stochastic Fokker- Planck Delay Differential Equations
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作者 Augustine O. Atonuje Jonathan Tsetimi 《Journal of Mathematics and System Science》 2016年第7期284-290,共7页
It is generally known that the solutions of deterministic and stochastic differential equations (SDEs) usually grow linearly at such a rate that they may become unbounded after a small lapse of time and may eventual... It is generally known that the solutions of deterministic and stochastic differential equations (SDEs) usually grow linearly at such a rate that they may become unbounded after a small lapse of time and may eventually blow up or explode in finite time. If the drift and diffusion functions are globally Lipschitz, linear growth may still be experienced, as well as a possible blow-up of solutions in finite time. In this paper, a nonlinear scalar delay differential equation with a constant time lag is perturbed by a multiplicative Ito-type time - space white noise to form a stochastic Fokker-Planck delay differential equation. It is established that no explosion is possible in the presence of any intrinsically slow time - space white noise of Ito - type as manifested in the resulting stochastic Fokker- Planck delay differential equation. Time - space white noise has a role to play since the solution of the classical nonlinear equation without it still exhibits explosion. 展开更多
关键词 Explosion non-linear stochastic Fokker Planck delay differential equation time - space white noise finite time.
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<i>L</i>-Stable Block Hybrid Second Derivative Algorithm for Parabolic Partial Differential Equations
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作者 Fidele Fouogang Ngwane Samuel Nemsefor Jator 《American Journal of Computational Mathematics》 2014年第2期87-92,共6页
An L-stable block method based on hybrid second derivative algorithm (BHSDA) is provided by a continuous second derivative method that is defined for all values of the independent variable and applied to parabolic par... An L-stable block method based on hybrid second derivative algorithm (BHSDA) is provided by a continuous second derivative method that is defined for all values of the independent variable and applied to parabolic partial differential equations (PDEs). The use of the BHSDA to solve PDEs is facilitated by the method of lines which involves making an approximation to the space derivatives, and hence reducing the problem to that of solving a time-dependent system of first order initial value ordinary differential equations. The stability properties of the method is examined and some numerical results presented. 展开更多
关键词 HYBRID Second DERIVATIVE Method Off-step Point PARABOLIC Partial differential equations
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Differential Transform Method for Some Delay Differential Equations
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作者 Baoqing Liu Xiaojian Zhou Qikui Du 《Applied Mathematics》 2015年第3期585-593,共9页
This paper concentrates on the differential transform method (DTM) to solve some delay differential equations (DDEs). Based on the method of steps for DDEs and using the computer algebra system Mathematica, we success... This paper concentrates on the differential transform method (DTM) to solve some delay differential equations (DDEs). Based on the method of steps for DDEs and using the computer algebra system Mathematica, we successfully apply DTM to find the analytic solution to some DDEs, including a neural delay differential equation. The results confirm the feasibility and efficiency of DTM. 展开更多
关键词 differential Transform METHOD Delay differential equation METHOD of stepS ANALYTIC SOLUTION Approximate SOLUTION
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求解随机微分方程split-step欧拉方法的收敛性
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作者 贾俊梅 《烟台大学学报(自然科学与工程版)》 CAS 2014年第2期90-94,共5页
给出随机微分方程的split-step欧拉格式的算法,并证明了当方程的偏移系数和扩散系数均满足线性增长条件和李普希兹条件的情况下,此方法用以求解随机微分方程的收敛性,并且求出强收敛的阶是1/2.同时证明了split-step近似解的均方收敛理论.
关键词 随机微分方程 split-step欧拉方法 收敛性
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基于RKADE-SS-FDTD方法的无条件稳定高阶CFS-PML算法
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作者 李建雄 魏之栋 《天津工业大学学报》 CAS 北大核心 2023年第2期55-60,共6页
为了提升完全匹配层(PML)算法的吸收效果,提出了一种用于截断非磁化等离子体的高阶复频率偏移PML(HO-CFS-PML)算法。该算法将龙格-库塔辅助微分方程(RKADE)算法和分裂步时域有限差分(SS-FDTD)算法相结合,来对非磁化等离子体进行仿真。... 为了提升完全匹配层(PML)算法的吸收效果,提出了一种用于截断非磁化等离子体的高阶复频率偏移PML(HO-CFS-PML)算法。该算法将龙格-库塔辅助微分方程(RKADE)算法和分裂步时域有限差分(SS-FDTD)算法相结合,来对非磁化等离子体进行仿真。在计算域边界处,引入了由辅助微分方程(ADE)算法实现的HO-CFS-PML用于截断开放区域中的等离子体,并通过一个数值算例进行模拟。结果表明:所提出的HO-RKADE-SS-CFS-PML算法在柯朗-弗里德里希斯-列维(CFL)数(CFLN)为8时,相比于传统CFS-PML算法,时间减少率可以达到79.2%,证明了该算法能够有效地消除柯朗-弗里德里希斯-列维(CFL)稳定性条件的约束,大大节省了计算时间;此外,HO-RKADE-SS-CFS-PML算法的最大相对反射误差可以达到-115 dB,明显小于其他算法的相对反射误差,并且随着CFLN的增大,该算法不会像RKADE-ADI-CFS-PML算法那样产生明显的误差增大现象,证明本文算法在大时间步长下拥有比其他算法更好的吸收电磁波的能力。 展开更多
关键词 高阶复频率偏移完全匹配层 分裂步时域有限差分法 龙格-库塔辅助微分方程 非磁化等离子体
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MEAN SQUARE STABILITY AND DISSIPATIVITY OF SPLIT- STEP THETA METHOD FOR NONLINEAR NEUTRAL STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH POISSON JUMPS 被引量:3
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作者 Haiyan Yuan Jihong Shen Cheng Song 《Journal of Computational Mathematics》 SCIE CSCD 2017年第6期766-779,共14页
In this paper, a split-step 0 (SST) method is introduced and used to solve the non- linear neutral stochastic differential delay equations with Poisson jumps (NSDDEwPJ). The mean square asymptotic stability of the... In this paper, a split-step 0 (SST) method is introduced and used to solve the non- linear neutral stochastic differential delay equations with Poisson jumps (NSDDEwPJ). The mean square asymptotic stability of the SST method for nonlinear neutral stochastic differential equations with Poisson jumps is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the SST method with ∈ E (0, 2 -√2) is asymptotically mean square stable for all positive step sizes, and the SST method with ∈ E (2 -√2, 1) is asymptotically mean square stable for some step sizes. It is also proved in this paper that the SST method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved. 展开更多
关键词 Neutral stochastic delay differential equations Split-step method Stability Poisson jumps.
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Computable extensions of generalized fractional kinetic equations in astrophysics 被引量:1
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作者 Vinod Behari Lal Chaurasia Shared Chander Pandey 《Research in Astronomy and Astrophysics》 SCIE CAS CSCD 2010年第1期22-32,共11页
Fractional calculus and special functions have contributed a lot to mathematical physics and its various branches. The great use of mathematical physics in distinguished astrophysical problems has attracted astronomer... Fractional calculus and special functions have contributed a lot to mathematical physics and its various branches. The great use of mathematical physics in distinguished astrophysical problems has attracted astronomers and physicists to pay more attention to available mathematical tools that can be widely used in solving several problems of astrophysics/physics. In view of the great importance and usefulness of kinetic equations in certain astrophysical problems, the authors derive a generalized fractional kinetic equation involving the Lorenzo-Hartley function, a generalized function for fractional calculus. The fractional kinetic equation discussed here can be used to investigate a wide class of known (and possibly also new) fractional kinetic equations, hitherto scattered in the literature. A compact and easily computable solution is established in terms of the Lorenzo-Hartley function. Special cases, involving the generalized Mittag-Leffler function and the R-function, are considered. The obtained results imply the known results more precisely. 展开更多
关键词 fractional differential equations - Mittag-Leffler functions - reaction- diffusion problems - Lorenzo-Hartley function
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A CLASS OF TWO-STEP CONTINUITY RUNGE-KUTTA METHODS FOR SOLVING SINGULAR DELAY DIFFERENTIAL EQUATIONS AND ITS STABILITY ANALYSIS 被引量:1
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作者 Xin Leng De-gui Liu +1 位作者 Xiao-qiu Song Li-rong Chen 《Journal of Computational Mathematics》 SCIE EI CSCD 2005年第6期647-656,共10页
In this paper, a class of two-step continuity Runge-Kutta(TSCRK) methods for solving singular delay differential equations(DDEs) is presented. Analysis of numerical stability of this methods is given. We consider ... In this paper, a class of two-step continuity Runge-Kutta(TSCRK) methods for solving singular delay differential equations(DDEs) is presented. Analysis of numerical stability of this methods is given. We consider the two distinct cases: (i)τ≥ h, (ii)τ 〈 h, where the delay τ and step size h of the two-step continuity Runge-Kutta methods are both constant. The absolute stability regions of some methods are plotted and numerical examples show the efficiency of the method. 展开更多
关键词 Analysis of numerical stability Singular delay differential equations Two-step continuity Runge-Kutta methods
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Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation
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作者 Lei Zhang Chaofeng Zhang Mengya Liu 《International Journal of Modeling, Simulation, and Scientific Computing》 EI 2020年第1期42-57,共16页
According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial,a variable-order and variable-step-size numeric... According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial,a variable-order and variable-step-size numerical method for solving differential equations is designed.The stability properties of the formulas are discussed and the stability regions are analyzed.The deduced methods are applied to a simulation problem.The results show that the numerical method can satisfy calculation accuracy,reduce the number of calculation steps and accelerate calculation speed. 展开更多
关键词 Numerical method variable step size variable order hermite interpolation ordinary differential equations
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A Comparative Analysis of the New -3(-n) - 1 Remer Conjecture and a Proof of the 3n + 1 Collatz Conjecture
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作者 Mike Remer 《Journal of Applied Mathematics and Physics》 2023年第8期2216-2220,共5页
This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An... This scientific paper is a comparative analysis of two mathematical conjectures. The newly proposed -3(-n) - 1 Remer conjecture and how it is related to and a proof of the more well known 3n + 1 Collatz conjecture. An overview of both conjectures and their respective iterative processes will be presented. Showcasing their unique properties and behavior to each other. Through a detailed comparison, we highlight the similarities and differences between these two conjectures and discuss their significance in the field of mathematics. And how they prove each other to be true. 展开更多
关键词 -3(-n) - 1 Remer Conjecture 3n + 1 Collatz Conjecture Comparative Analysis PROOF Natural Numbers Integer Sequences Factorial Processes Par-tial differential equations Bounded Values Collatz Conjecture Collatz Algo-rithm Collatz Operator Collatz Compliance And Mathematical Conjectures
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A SPARSE-GRID METHOD FOR MULTI-DIMENSIONAL BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS 被引量:2
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作者 Guannan Zhang Max Gunzburger Weidong Zhao 《Journal of Computational Mathematics》 SCIE CSCD 2013年第3期221-248,共28页
A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e.... A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathe- matical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and (adaptive) hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme. 展开更多
关键词 Backward stochastic differential equations Multi-step scheme Gauss-Hermite quadrature rule Adaptive hierarchical basis Sparse grids.
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延迟微分方程隐-显式线性多步法的稳定性 被引量:1
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作者 袁海燕 赵景军 《系统仿真学报》 CAS CSCD 北大核心 2009年第23期7418-7420,7427,共4页
通过研究延迟微分方程隐-显式线性多步法的稳定性,给出两类特殊的隐-显式方法即隐-显式Euler方法和隐-显式BDF方法的稳定性结论,证明了隐-显式Euler方法是P-稳定的,隐-显式BDF方法不是P-稳定的。为了克服边界轨迹法刻画复空间稳定区域... 通过研究延迟微分方程隐-显式线性多步法的稳定性,给出两类特殊的隐-显式方法即隐-显式Euler方法和隐-显式BDF方法的稳定性结论,证明了隐-显式Euler方法是P-稳定的,隐-显式BDF方法不是P-稳定的。为了克服边界轨迹法刻画复空间稳定区域的困难,给出了一种新的复空间上稳定区域的刻画方法,并用这种方法给出了隐-显式BDF方法的数值稳定性区域的描述,最后通过数值算例验证了这种刻画稳定区域的方法的可行性。 展开更多
关键词 延迟微分方程 -显式线性多步法 稳定性 稳定区域
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两步Runge-Kutta法求解延迟微分方程的GPL_m-稳定性(英文) 被引量:1
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作者 丛玉豪 蒋成香 《系统仿真学报》 CAS CSCD 北大核心 2011年第7期1366-1368,共3页
讨论了两步Runge-Kutta方法求解延迟微分方程的数值稳定性,分析了求解线性试验方程的两步Runge-Kutta方法的稳定性态。证明了两步Runge-Kutta方法是GPLm-稳定的,当且仅当它求解常微分方程是L-稳定的。
关键词 延迟微分方程 两步Runge-Kutta方法 GPL-稳定性 L-稳定性
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四阶Runge-Kutta算法的优化分析 被引量:9
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作者 刘利斌 王伟 《成都大学学报(自然科学版)》 2007年第1期19-21,共3页
引用最优化思想和随机搜索技术,对一般的四阶Runge-Kutta算法进行分析,得出一类精度较高的算法,这些算法能满足我们预先给定精度,并且其局部截断误差为O(h5).
关键词 常微分方程初值问题 最优化 随机搜索 四阶Runge—Kutta法
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中立型多延迟积分微分代数方程二步Runge-Kutta方法渐近稳定性(英文)
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作者 袁海燕 宋成 +1 位作者 赵景军 曲绍平 《黑龙江大学自然科学学报》 CAS 北大核心 2013年第2期187-194,共8页
分析向量值形式的中立型多延迟积分微分代数方程二步Runge-Kutta方法的渐近稳定性。首先给出中立型多延迟积分微分代数方程解析解渐近稳定的定义,并给出使得解析解渐近稳定的充分条件。随后给出二步Runge-Kutta方法的一般形式和数值解... 分析向量值形式的中立型多延迟积分微分代数方程二步Runge-Kutta方法的渐近稳定性。首先给出中立型多延迟积分微分代数方程解析解渐近稳定的定义,并给出使得解析解渐近稳定的充分条件。随后给出二步Runge-Kutta方法的一般形式和数值解渐近稳定的定义,给出数值方法渐近稳定的充分条件,最后证明A-稳定的二步Runge-Kutta方法求解中立型多延迟积分微分代数方程是渐近稳定的,并给出数值算例验证结论。 展开更多
关键词 渐近稳定性 多延迟积分微分代数方程 二步Runge—Kutta方法 A-稳定 特征多项式 数值方法
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