Rational Solutions of First Order Algebraic Ordinary Differential Equations

Let f(t,y,y’)=∑_i=0ⁿ a_i(t,y)y’ⁱ=0 be an irreducible first order ordinary differential equation with polynomial coefficients.Eremenko in 1998 proved that there exists a constant C such that every rational solution of f(t,y,y’)=0 is of degree not greater than C.Examples show that this degree bound C depends not only on the degrees of f in t,y,y’ but also on the coefficients of f viewed as the polynomial in t,y,y’.In this paper,the authors show that if f s...     

Descent of Ordinary Differential Equations with Rational General Solutions

Journal of Systems Science and Complexity - Let F be an irreducible differential polynomial over k(t) with k being an algebraically closed field of characteristic zero. The authors prove that F = 0...     

Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero

This paper considers the class of autonomous algebraic ordinary differential equations(AODEs) of order one, and studies their Liouvillian general solutions. In particular, let F(y, w) = 0 be a rational algebraic curve over C. The authors give necessary and sufficient conditions for the autonomous first-order AODE F(y, y′) = 0 to have a Liouvillian solution over C. Moreover, the authors show that a Liouvillian solution α of this equation is either an algebraic function over C(x) or an algebraic f...     

常微分方程与代数方程求根

讨论了常微分方程与代数方程求根之间的联系。建立了常微分方程的解与代数方程根的一种联系关系，通过这种联系借助于常微分方程的求解，给出以大范围的初值求出代数方程全部实根的一种方法。还讨论了这种方法的几种具体数值求根算法，并对其中的一些有意义的现象进行了分析。     

Parallel Rosenbrock Methods for Differential Algebraic Equations

1. INTRODUCTIONWe consider the system of dtherelltial-algebraic equations(DAE)where y is in some space E and z in EI. For convenience of notations we assume that y and z are scalarqualities. All subsequeDt formulas remain valid for vectors if the derivatives are interpreted as multilinearmappings. We suppose that the initial values are consistent, i.e. g(yo, to) = 0. Moreover j and g are assumedto be sufficiently differentiable. We also assume the DAE system to be of index one; it me…     

IMPLICITIZATION OF PARTIAL DIFFERENTIAL RATIONAL PARAMETRIC EQUATIONS

In this paper, we propose algorithms for the following problems in the implicitization of a set of partial differential rational parametric equations P. （1）To find a characteristic set for the implicit prime ideal of P; （2） To find a canonical representation for the image of P; （3）To decide whether the parameters of P are independent, and if not, to re-parameterize P so that the new parametric equations have independent parameters; （4） To compute the inversion maps of P, and as a consequence, to decide whether P is proper.     

一种基于量子化状态系统的刚性ODE求解方法

量子化状态系统(Quantized State System,QSS)在求解一般常微分方程(Ordinary Differential Equation,ODE)系统时,比传统基于时间离散的积分方法更具优势,但QSS方法不适合求解刚性ODE系统,为此提出一种基于量子化状态系统的步进校正优化算法(Step-correction Optimization Algorithm Based on QSS,SCOA based-on QSS),它结合QSS方法及隐式算法中梯形积分法的思想,以有效提高刚性ODE系统的求解精度和效率。通过对3个典型刚性ODE算例的仿真求解,结果表明,SCOA based-on QSS算法总体上比其他算法更具优势,同时在适当减小量子大小时能显著提高仿真精度。     

一类分数阶常微分方程的显式算法

利用Caputo导数的性质和求解普通积分方程的Admas技巧,得到一种求解一类分数阶微分方程初值问题的显式方法;证明了其整体误差估计为O(hθ),θ=min{α,2},并进行了稳定性分析.该方法在α>1时比已有的几种显式方法具有更高的数值精度和收敛阶.     

On the Meromorphic Solutions of Linear Differential Equations

In this paper, we investigate the growth of meromorphic solutions of higher order linear differential equation f^（k） ＋Ak-1 （z）e^Pk-1^（z） f^（k-1） ＋…＋A1 （z）e^P1（z） f′ ＋Ao（z）e^Po（z） f = 0 （k ≤ 2）, where Pj（z） （j = 0, 1,..., k - 1） are nonconstant polynomials such that deg Pj = n （j = 0, 1,..., k - 1） and Aj（z）（≠ 0） （j = 0, 1,..., k - 1） are meromorphic functions with order p（Aj） 〈 n （j = 0, 1,..., k - 1）.     

The Algebro-Geometric Method for Solving Algebraic Differential Equations——A Survey

Journal of Systems Science and Complexity - This paper presents the algebro-geometric method for computing explicit formula solutions for algebraic differential equations (ADEs). An algebraic...     

求解常微分方程的一类线性多步方法

本文给出常微分方程y′=f(t,y)及″=f(t,y)的一类线性多步方法,它们的系数容易在计算机上生成。     

Some Techniques for Solving Ordinary Differential Equations with Discontinuities in Real－time Simulation

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基于思维进化算法的常微分方程组演化建模

在基本思维进化算法（MindEvolutionaryComputation，原名Mind-Evolution-BasedMachineLearning）框架的基础上，吸取遗传规划（GP）中的树形编码思想，提出了利用树结构进行信息抽取的方法，进而实现了用于常微分方程组演化建模的趋同、异化算子，并获取了优良的效果，使MEC在非数值优化领域中得到了进一步应用。最后的仿真实例的结果表明，同GP方法比较，MEC方法具有计算速度快、建模结果优和全局搜索性能好等明显优点。     

常微分方程初值问题并行算法研究现状

本文对常微分方程初值问题数值求解的并行算法进行综述,给出并行算法的应用前景和构造的一些途径,同时指出并行化的主要困难和一些解决的方法。     

Distributed Solving Linear Algebraic Equations with Switched Fractional Order Dynamics

This paper proposes a novel distributed optimization algorithm with fractional order dynamics to solve linear algebraic equations. Firstly, the authors proposed “Consensus + Projection” flow with fractional order dynamics, which has more design freedom and the potential to obtain a better convergent performance than that of conventional first order algorithms. Moreover, the authors prove that the proposed algorithm is convergent under certain iteration order and step-size. Furthermore,the author...     

一类微分代数系统并行仿真算法

利用对误差的预校正技术,本文提出一类微分代数系统的并行仿真算法,进行仿真误差估计。对于一个模型问题进行稳定性分析,画出稳定区域。计算实例表明算法是有效的。     

改进的向后微分公式

采用增加方法步数的办法构造了两类求解刚性微分方程的改进的向后微分公式IBDF1及IBDF2。理论分析和数值试验表明,这两类新方法在较大地改善了BDF的稳定性的同时保持了BDF原有的主要优点。     

微分代数控制问题的数值计算方法

描述许多轨道控制问题的方程通常构成非线性半显式的微分代数系统。本文提出一些数值方法来计算这些控制问题的控制规律。对于一个模型问题,本文进行了稳定性分析,并且给出了稳定性区域。文中估计了这些方法的全局误差,它们给出控制误差与计算步长的关系。     

基于局部微分方程组的生态系统模拟

将变系数微分方程组描述的一个生态系统模型,分解为几个常系数微分方程组描述的子模型。调用Matlab中的odes函数,通过研究这些局部常系数微分方程组而实现对整个生态系统的模拟与仿真。运行结果表明,模型的拟合性教好;抗性分析表明,计算结果可信。因此这种仿真算法在生态系统模拟中可行。     

非线性中立型延迟微分方程的散逸性

主要研究非线性中立型延迟微分方程本身及其数值方法的散逸性问题。首先,对此类中立型延迟微分方程理论解的散逸性给出了充分条件;随后,应用一类线性多步法求解至该类问题,证明了在适当条件下,其数值解也具有散逸性;最后,数值试验进一步验证了理论结果的正确性。