基于LS-SVM求解非线性常微分方程组的近似解

提出了一种基于最小二乘支持向量机(LS-SVM)的改进方法求解非线性常微分方程组初值问题的近似解.利用径向基核函数(RBF)可导的特点对LS-SVM模型进行改进,将含核函数导数形式的LS-SVM模型转化为优化问题进行求解.方法可在原始对偶集中获得近似解的最佳表示,所得近似解连续可微,且精度较高.给出数值算例,通过与真实解的对比验证了所提方法的准确性和有效性.     

具有π-导数模糊微分方程的近似解

在模糊值函数具有π-导数意义下研究一阶模糊微分方程的模糊初值问题,将模糊微分方程转化成同解的常微分方程,利用变分迭代算法给出方程的近似解,给出了具体算例。     

基于同伦分析法的轨道电路暂态仿真分析北大核心CSCD

为了进行轨道电路暂态分析,采用同伦分析法对轨道电路传输线方程求解。将Caputo分数阶导数作为线性算子,建立轨道电路传输线方程的高阶形变方程,求解得轨道电路传输线方程三阶近似解。根据不同的道床电阻确定其收敛区域,对轨道电路电压波暂态响应进行仿真分析。仿真结果表明,将分数阶导数作为线性算子的同伦分析法可以准确分析轨道电路传输线电压波传输特征和过程,为轨道电路暂态分析提供了一种新的方法。     

微分积分方程 一阶拟线性偏微分方程組初值問题近似解的差分方法

W.Urich描述了一个差分方法,它用于近似求解含有m个方程、m个未知函数及n+1个自变量的一阶拟线性偏微分方程组的初值问题。这方法是根据这样一种事实:在每一个方程中,同一个函数的偏导数定义了一个方向导     

基于场方法的非线性系统求解

用场方法联立多尺度法求单自由度的非线性系统的近似解．将两个状态方程的一个状态变量看作是另一个状态变量和时间的一个场函数,把原系统化为求解具有初始条件的基本方程,通过多尺度展开,逐个摄动方程求解,获得了振幅和相位的一阶近似微分方程,作为例子,求得了非线性振动系统的一阶近似解,并和数值解进行比较,两者吻合较好。     

一类强非线性二阶微分方程的多模态近似解析解研究

利用自治力学系统的哈密顿函数为守恒量的性质,提出一种求非线性二阶微分方程多模态近似解析解的方法,称为哈密顿函数法.首先,介绍哈密顿函数法求多模态近似解的基本理论.其次,以质点在旋转的抛物线上运动为模型建立强非线性二阶微分方程.最后,用哈密顿函数法求得在给定初始条件和参数下强非线性二阶微分方程的三模态近似解析解表达式,作出三模态近似解析解的解曲线,并与直接用Mathematica软件作出的解曲线进行比较,讨论三模态近似解析解的精确性.结果表明：用哈密顿函数法求得的三模态近似解析解的解曲线与直接用Mathematica软件作出的解曲线十分吻合.     

大型群体多属性决策的目标函数线性回归法

在定义标准决策值的基础上,运用线性回归的方法建立了带有约束条件的大型群体多属性决策目标函数线性回归模型,证明了该模型解的存在性和唯一性.大型群体多属性决策目标函数线性回归法对各专家的标准决策值的信息进行了最优意义上的集结和协调.此外,对大型群体多属性决策目标函数线性回归模型进行了讨论,并获得了一整套可用Matlab软件求解该模型最优解的算法,具体应用算例验证了所提算法的有效性.     

基于GEP算法的高阶常微分方程预测模型

针对动态系统预测建模中建模效率低,无显式模型的缺陷。提出一种基于基因表达式编程（GEP）的高阶常微分方程预测模型（GEP-HODE）。将一维数据的变化特性使用高阶微分进行表示,通过GEP对高阶微分数据进行建模,得到显式模型。对高阶常微分方程模型进行降阶处理,使用数值方法进行求解,得到预测值。该方法利用了GEP算法“基因型-表现型”的编码特性,实现了模型建立与参数优化的同步,大幅度提升建模效率。以太阳黑子年平均数作为实验数据建模预测,结果表明,该方法相比GP混合建模方法有更高的效率,相比混合BP神经网络模型等方法有更好的精度。     

基于δ法求解高阶时变非线性微分方程

该文基于二阶线性微分方程δ法绘制相平面原理,提出一种新颖而简单计算圆弧圆心和半径的方法实现高阶时变非线性微分方程相平面的作图,从而得到求解高阶时变非线性微分方程时域解的算法,并与龙格-库塔法等解析法相比具有计算简单、结果精度高的特点。     

一种高阶无迹卡尔曼滤波方法

现有的研究中,高阶无迹变换（Unscented transform,UT）还不存在具体的解析解,因此,无法利用高阶无迹变换获得具备更高精度的高阶无迹卡尔曼滤波器（Unscented Kalman filter,UKF）.为了解决这一问题,本文在五阶容积变换（Cubature transform,CT）的基础上,通过引入一个自由参数κ,得到高阶无迹变换的解析解,从而获得了高阶无迹卡尔曼滤波器（Unscented Kalman filter,UKF）.同时验证了现有的五阶容积变换和五阶无迹变换分别是本文所提出的高阶无迹变换在κ=2和κ=6-n时的两个特例.进而分析和讨论了高阶无迹卡尔曼滤波器在系统不同维数条件下κ值的最优选取,并讨论了其稳定性.纯方位跟踪模型和弹道目标再入模型仿真验证了本文方法的正确性,且与现有方法相比具有更高的精度.     

Using the iterative reproducing kernel method for solving a class of nonlinear fractional differential equations

In previous works, we have devoted to using the reproducing kernel methods solving integer order differential equations, based on the review of previous works, in this paper, we mainly present a method for solving a class of higher order fractional differential equations with general boundary value problems by using Taylor formula into reproducing kernel space. Its analytical solution is represented in the form of series. The analytical solution and approximate solution obtained by this method is given and it is uniformly converge to the exact solution and its corresponding derivatives. The numerical examples are studied to demonstrate the accuracy of the present method.     

An alternative use of fuzzy transform with application to a class of delay differential equations

Fuzzy transforms (or F-transforms for short) are an approximation technique recently introduced. The main application is referred to image and data compression. There are really few works devoted to the use of F-transform for solving ordinary differential equations. In the present paper, an F-transform-based Picard-like numerical scheme is proposed in order to solve a class of delay differential equations. For linear cases, the proposed approach leads to a non-recursive approximate solution by means of operational matrices and vectors of known quantities. Numerical results show the good performance of the proposed method against known solutions.     

面向调控网络参数学习的无迹粒子滤波算法

目前基于微分方程模型学习网络参数的工作普遍基于卡尔曼滤波器,对所分析系统有线性假设前提,而基因调控网络具有强非线性,因此需要更适用于非线性模型的方法。提出了一种基于无迹粒子滤波器学习基因调控网络参数的方法,由于粒子滤波方法不受模型线性假设的约束,因此能够对非线性系统进行更好的拟合。通过对Repressillar模型中隐变量与未知参数的估计并与无迹卡尔曼滤波器所获结果的比较,提出的算法有效减少了估计误差。并对粒子数目对结果的影响进行了分析。相较于卡尔曼滤波器,无迹粒子滤波方法对于调控网络参数学习精度更高。粒子数目太少或太多都会减弱估计精度,因此选择适当的粒子数目非常重要。     

Generalized newton multi-step iterative methods GMNp,m for solving system of nonlinear equations

A generalization of the Newton multi-step iterative method is presented, in the form of distinct families of methods depending on proper parameters. The proposed generalization of the Newton multi-step consists of two parts, namely the base method and the multi-step part. The multi-step part requires a single evaluation of function per step. During the multi-step phase, we have to solve systems of linear equations whose coefficient matrix is the Jacobian evaluated at the initial guess. The direct inversion of the Jacobian it is an expensive operation, and hence, for moderately large systems, the lower-upper triangular factorization (LU) is a reasonable choice. Once we have the LU factors of the Jacobian, starting from the base method, we only solve systems of lower and upper triangular matrices that are in fact computationally economical. The developed families involve unknown parameters, and we are interested in setting them with the goal of maximizing the convergence order of the global method. Few families are investigated in some detail. The validity and numerical accuracy of the solution of the system of nonlinear equations are presented via numerical simulations, also involving examples coming from standard approximations of ordinary differential and partial differential nonlinear equations. The obtained results show the efficiency of constructed iterative methods, under the assumption of smoothness of the nonlinear function.     

Analysis of systems of singular ordinary differential equations

Systems of ordinary differential equations with a small parameter at the derivative and specific features of the construction of their periodic solution are considered. Sufficient conditions of existence and uniqueness of the periodic solution are presented. An iterative procedure of construction of the steady-state solution of a system of differential equations with a small parameter at the derivative is proposed. This procedure is reduced to the solution of a system of nonlinear algebraic equations and does not involve the integration of the system of differential equations. Problems of numerical calculation of the solution are considered based on the procedure proposed. Some sources of its divergence are found, and the sufficient conditions of its convergence are obtained. The results of numerical experiments are presented and compared with theoretical ones. Translated from Kibemetika i Sistemnyi Analiz, No. 5, pp. 103–110, September–October, 1999.     

Multiplicative stability criteria based on difference solutions of ordinary differential equations

For computer analysis of Lyapunov stability, multiplicative criteria are proposed that are based on difference approximations to solutions of the Cauchy problem. These criteria can be applied to ordinary differential equations in normal form and include the necessary and sufficient stability conditions. For a system of linear equations with constant coefficients, information on the characteristic polynomial of the coefficient matrix and its roots is not used. The stability analysis is combined with difference solution and simulation of error accumulation. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 127–142, January–February 2006.     

The use of polynomial spline functions for the solution of system of second order delay differential equations

This paper presents the use of spline functions of polynomial form to approximate the solution of system of second order delay differential equations. The error analysis and stability of the method are theoretically investigated. A numerical example is given to illustrate the applicability, accuracy and stability of the proposed method.     

Stability of 2h-step spline method for delay differential equations

In this paper we consider a linear test equation to study the stability analysis of 2h-step spline method for the solution of delay differential equations. We prove that, this method is P-stable for cubic spline.     

Analytical solution for a class of linear quadratic open-loop Nash game with multiple players

In this paper, the Nash equilibria for differential games with multiple players is studied. A method for solving the Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic game with multiple players is presented and analytical solution is given for a type of differential games in which the system matrix can be diagonalizable. As the special cases, the Nash equilibria for some type of differential games with particular structure is studied also, and some results in previous literatures are extended. Finally, a numerical example is given to illustrate the effectiveness of the solution procedure.     

Analytical solution for a class of linear quadratic open-loop Nash game with multiple players

In this paper, the Nash equilibria for differential games with multiple players is studied. A method for solving the Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic game with multiple players is presented and analytical solution is given for a type of differential games in which the system matrix can be diagonalizable. As the special cases, the Nash equilibria for some type of differential games with particular structure is studied also, and some results in previous literatures are extended. Finally, a numerical example is given to illustrate the effectiveness of the solution procedure.