退化时滞中立型微分系统解的存在唯一性及指数估计

本文主要讨论退化时滞中立型微分系统解的存在唯一性及指数估计问题.通过定义正则矩阵对讨论退化时滞中立型微分系统解的存在唯一性.再定义基解矩阵以及Laplace变换,给出该系统的通解表达式,最后利用通解表达式和Gronwall-Bellman积分不等式给出该系统解的指数估计及解的精确指数界限.     

一类具有转移条件高阶微分算子的J-自伴性

本文研究了一类具有转移条件的高阶复系数微分算子的J-自伴性,利用J-对称微分算式的拉格朗日双线性型、J-自伴算子的定义及矩阵表示的方法,证明了这类微分算子是J-自伴的,且对应于不同特征值的特征向量和特征子空间都是C-正交的.     

半张量积下矩阵方程组AX=B,XC=D的最小二乘解

本文研究了半张量积下矩阵方程组AX=B,XC=D在不同情况下的最小二乘解X*∈R~(p×q),其中矩阵A∈R~(m×n),B∈R~(h×k),C∈R~(a×b),D∈R~(l×d)给定.根据半张量积的定义将其转变为普通乘积下的矩阵方程组,再结合矩阵奇异值分解及矩阵微分给出该方程组在不同情况下最小二乘解的解析表达式,并用数值算例加以验证.     

具有转移条件的四阶微分算子自共轭边界条件的统一规范型

研究了具有转移条件的四阶正则微分算子自共轭边界条件的统一规范型.在标准型的基础上通过对自共轭边界条件矩阵左乘非奇异矩阵和右乘辛矩阵给出了四阶微分算子自共轭边界条件的统一规范型.结果表明具有转移条件的四阶自共轭微分算子的边界条件的统一规范型不仅与边界条件矩阵的秩有关,而且与转移条件矩阵的行列式有关.     

分数阶一般退化微分系统的通解

本文研究了系数矩阵不是方阵情形的分数阶一般退化微分系统的解.通过定义可解阵对,获得分数阶一般退化微分系统的通解表达式.该结果推广了整数阶退化微分系统和分数阶常微分系统解的相应结论.     

辛群的ω子集的微分结构

对复平面中单位圆周上的ω研究了Sp(2n)中以ω为特征值的矩阵所形成的子集的微分结构,据此定义了由复平面之单位圆周上的所有ω所参数化的ω指标理论.     

微分概念的历史发展及教学启示

微分的形式化定义是学生学习微分概念的主要困难.微分概念的历史发展表明,形式化的微分定义是微积分严格化的产物,朴素的微分定义更能体现微积分思想,而非标准分析给微分概念带来重生.在微积分学中应用非形式化的方法构建微分概念,以微分为主线(传统教材一般以导数为主线)进行微积分教学可以促进学生学习效果.     

二阶非线性矩阵微分系统的振动判据

考虑二阶非线性矩阵微分系统t∈[a,+∞)a≥0,其中关于R、Q、Y、F的基本假设和解的振动定义及有关记号如文[4]引言所述,记n阶实对称矩阵的全体为S~(n×n),A_k(a_(ij))_(k×k),容易证明以下结论,其中A,B∈S~(n×n)。     

系数在结合超代数的矩阵李超代数的2-上循环

设α是域F上的结合超代数满足[α,α]=a或a=F.证明了当m+n＞1时,H2(glm|n(α),F)≌HC1(a,F).定义了一大类广义微分算子李超代数,作为W-无穷代数W∞(glN)的推广.确定了这些李超代数的2-上循环.同时给出了矩阵量子微分算子李超代数的2-上同调群.     

系数在结合超代数的矩阵李超代数的2-上循环

设α是域F上的结合超代数满足[α,α]=α或α=F．证明了当m n>1时,H2(glm|n(α),F)(?)HC1(α,F)．定义了一大类广义微分算子李超代数,作为W-无穷代数W∞(glN)的推广．确定了这些李超代数的2-上循环．同时给出了矩阵量子微分算子李超代数的2-上同调群．     

J-selfadjointness of a class of high-order differential operators with transmission conditions

This paper is to investigate the J-selfadjointness of a class of high-order complex coefficients differential operators with transmission conditions. Using the Lagrange bilinear form of J-symmetric differential equations, the definition of J-selfadjoint differential operators and the method of matrix representation, we prove that the operator is J-selfadjoint operator, and the eigenvectors and eigen-subspaces corresponding to different eigenvalues are C-orthogonal.     

一类含无界时滞的非线性系统的渐近稳定性

基于泛函微分方程的稳定性理论,首先通过构造Lyapunov泛函,再利用矩阵不等式的性质和范数的定义判定矩阵的正定性和负定性.对一类含无界时滞的非线性系统的渐近稳定性问题,作进一步的探讨.     

Index-aware model order reduction for differential-algebraic equations

We introduce a model order reduction (MOR) procedure for differential-algebraic equations, which is based on the intrinsic differential equation contained in the starting system and on the remaining algebraic constraints. The decoupling procedure in differential and algebraic part is based on the projector and matrix chain which leads to the definition of tractability index. The differential part can be reduced by using any MOR method, we use Krylov-based projection methods to illustrate our approach. The reduction on the differential part induces a reduction on the algebraic part. In this paper, we present the method for index-1 differential-algebraic equations. We implement numerically this procedure and show numerical evidence of its validity.     

分数阶常微分方程的改进精细积分法

基于Mittag-Leffler函数的定义式,构造Mittag-Leffler矩阵函数的精细迭代计算格式.与常规指数函数的迭代格式相比,迭代递推中多了修正项,其表达式与分数阶导数的阶次有关.对于以Caputo分数导数定义的动力学分数阶常微分方程,使用基于Mittag-Leffler函数的精细积分法可计算方程解在各时间段端点对应函数值.算例表明了所提计算方法的有效性,其精度可由所增加修正项的阶次控制.     

The solvability of the initial problem for a degenerate linear hybrid system with variable coefficients

We consider a linear hybrid system with variable coefficients and known mode switching moments under the assumption that matrices at the derivative of the desired vector function are identically degenerate. We obtain the necessary and sufficient conditions for the existence of a piecewise smooth solution (either continuous or not in its definition domain) for the initial problem. We study an equivalent form of a nonstationary system of linear ordinary differential equations that is not resolved with respect to the derivative and is identically degenerate in its definition domain. We propose a constructive algorithm for obtaining such a form even if the rank of the matrix at the derivative is not constant.     

An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations

This paper aims to construct a general formulation for the Jacobi operational matrix of fractional integral operator. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, a reliable and efficient technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the Jacobi integral operational matrix to the fractional calculus. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Stability Analysis of Distributed-Order Hilfer–Prabhakar Systems Based on Inertia Theory

The notion of a distributed-order Hilfer–Prabhakar derivative is introduced, which reduces in special cases to the existing notions of fractional or distributed-order derivatives. The stability of two classes of distributed-order Hilfer–Prabhakar differential equations, which are generalizations of all distributed or fractional differential equations considered previously, is analyzed. Sufficient conditions for the asymptotic stability of these systems are obtained by using properties of generalized Mittag-Leffler functions, the final-value theorem, and the Laplace transform. Stability conditions for such systems are introduced by using a new definition of the inertia of a matrix with respect to the distributed-order Hilfer–Prabhakar derivative.     

To the question of the minimal number of inputs for linear differential algebraic systems

We examine a control linear system of ordinary differential equations with an identically degenerate matrix coefficient of the derivative of the unknown vector function. We study the question of the minimal dimension of the control vector when the system could be fully controllable on any segment in the domain of definition. The problem is investigated in the cases of stationary systems and the systems with real analytic and smooth coefficients for which some structural forms can be defined.     

On reflecting boundary conditions for space-fractional equations on a finite interval: Proof of the matrix transfer technique

Even in the one-dimensional case, dealing with the analysis of space-fractional differential equations on finite domains is a difficult issue. On a finite interval coupled with zero flux boundary conditions, different approaches have been proposed to define a space-fractional differential operator and to compute the solution to the corresponding fractional problem, but to the best of our knowledge, a clear relationship between these strategies is yet to be established. Here, by using the theory of α-stable symmetric Lévy flights and the master equation, we derive a discrete representation of the non-local operator embedding in its definition the concept of reflecting boundary conditions. We refer to this discrete operator as the reflection matrix and provide (and prove) a theorem on the analytic expression of its eigenvalues and eigenvectors. We then use this result to compare the reflection matrix to the discrete operator defined via the matrix transfer technique, and establish the validity of the latter technique in producing the correct solution to a space-fractional differential equation on a finite interval with reflecting boundary conditions. We finally discuss and emphasize the challenges in the generalisation of the proposed result to more than one spatial dimension.     

On the Cauchy problem for systems —Parabolic systems—

The author proposes the definition of thep-parabolic system, which is stable under the similar transformations, using thep-determinant of the matrix of differential operators. The relation between thep-parabolic systems andH ^∞ well-posedness is considered. To the memory of Lamberto Cattabriga