关于一类中立型偏微分方程

本文研究了一类一阶脉冲中立型偏泛函微分方程mild解的存在性和唯一性,改进并推广了文[11]的结果。     

New representation and factorizations of the higher-order ultraspherical-type differential equations

The paper deals with the class of linear differential equations of any even order 2α+4$2\alpha +4$, α∈_N0$\alpha \in {\mathbb{N}}_0$, which are associated with the so-called ultraspherical-type polynomials. These polynomials form an orthogonal system on the interval [−1,1]$[-1,1]$ with respect to the ultraspherical weight function (1−x²)^α${(1-x^2)}^{\alpha }$ and additional point masses of equal size at the two endpoints. The differential equations of “ultraspherical-type” were developed by R. Koekoek in 1994 by utilizing special function methods. In the present paper, a new and completely elementary representation of these higher-order differential equations is presented. This result is used to deduce the orthogonality relation of the ultraspherical-type polynomials directly from the differential equation property. Moreover, we introduce two types of factorizations of the corresponding differential operators of order 2α+4$2\alpha +4$ into a product of α+2$\alpha +2$ linear second-order operators.     

Farkas-type theorems for interval linear systems

We describe some Farkas-type necessary and sufficient conditions for solvability and feasibility of interval linear equations and inequalities, in a unified form. For the convenient choice in practice of theory and calculations, some different but equivalent forms of the Farkas-type conditions for interval linear systems are also discussed.     

On transformations z(t) = y(ϕ(t)) of ordinary differential equations

The paper describes the general form of an ordinary differential equation of the order n + 1 (n 1) which allows a nontrivial global transformation consisting of the change of the independent variable. A result given by J. Aczél is generalized. A functional equation of the form where are given functions, is solved on .     

Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation

In this paper, we present a power penalty function approach to the linear complementarity problem arising from pricing American options. The problem is first reformulated as a variational inequality problem; the resulting variational inequality problem is then transformed into a nonlinear parabolic partial differential equation (PDE) by adding a power penalty term. It is shown that the solution to the penalized equation converges to that of the variational inequality problem with an arbitrary order. This arbitrary-order convergence rate allows us to achieve the required accuracy of the solution with a small penalty parameter. A numerical scheme for solving the penalized nonlinear PDE is also proposed. Numerical results are given to illustrate the theoretical findings and to show the effectiveness and usefulness of the method. This work was partially supported by a research grant from the University of Western Australia and the Research Grant Council of Hong Kong, Grants PolyU BQ475 and PolyU BQ493.     

常系数线性方程组的一种新解法

对于常系数线性微分方程组:dx/dt=Ax(A是n阶实常数矩阵)通过特征根λ和对应的特征行向量K:K~T(A-λE)=0将微分方程组化为线性方程组:1°当有n个互异的特征根λ_1,λ_2,…,λ_n,对应的线性无关的特征行向量为K_1,K_2,…,K_n,若记K_i=(k_1,k_2,…,k_n)(i=1,2,…,n),则有方程组:(n∑i=1 k_ix_i)′=λ_j(n∑i=1 k_ix_I)(j=1,2,…,n);2°当有不同的特征根λ_1,λ_2,…,λ_m其重数分别为n_1,n_2,…,n_m,n_1+n_2+…+n_m=n,对应的线性无关的特征行向量为K_i=(k_1,K_2,…,k_n)(i=1,2,…,m),则有方程组:(n∑i=1 k_rx_r)′=λ_k(n∑i=1 k_rx_r)((A-λ_jE)x_(n_i)=0;i=1),(n∑i=1 k_rx_r)′=λ_j(n∑i=1k_rx_r)+c_(n_i)e~(λ_jt)((A-λ_kE)x_(i-1)=Ex_i,i=2,…,n_i).     

On the Growth of Components of Meromorphic Solutions of Systems of Complex Differential Equations

This paper investigates the problem of the growth of the components of meromorphic solutions of a class of a system of complex algebraic differential equations, and generalized some of N. Toda＇s results concerning the growth of differential equations to the case of systems of differential equations. The paper considers the existence of admissible solutions of the system of differential equations.     

Existence of solutions for impulsive partial neutral functional differential equations

This paper is concerned with partial neutral functional differential equations of first and second order with impulses. We establish some results of existence of mild solutions for these classes of equations.     

The calculus of the trigonometric functions

The trigonometric functions entered “analysis” when Isaac Newton derived the power series for the sine in his De Analysi of 1669. On the other hand, no textbook until 1748 dealt with the calculus of these functions. That is, in none of the dozen or so calculus texts written in England and the continent during the first half of the 18th century was there a treatment of the derivative and integral of the sine or cosine or any discussion of the periodicity or addition properties of these functions. This contrasts sharply with what occurred in the case of the exponential and logarithmic functions. We attempt here to explain why the trigonometric functions did not enter calculus until about 1739. In that year, however, Leonhard Euler invented this calculus. He was led to this invention by the need for the trigonometric functions as solutions of linear differential equations. In addition, his discovery of a general method for solving linear differential equations with constant coefficients was influenced by his knowledge that these functions must provide part of that solution.     

A short note on solvability of systems of interval linear equations

In this note we formulate necessary and sufficient conditions for strong solvability and feasibility of systems of linear interval equations in terms of absolute value inequalities.     

On Bounded Solutions of Systems of Linear Functional Differential Equations

Sufficient conditions of the existence and uniqueness of bounded on real axis solutions of systems of linear functional differential equations are established.     

Formulas for Liapunov Functions for Systems of Linear Difference Equations

Explicit quadratic Liapunov functions that provide necessaryand sufficient conditions for the asymptotic stability of thesystem of linear difference equations x (t + 1) = Ax(t) areconstructed by transforming the original systems to y (t + 1)= Gy(t), where G is a companion matrix associated with the characteristicpolynomial of A. A necessary and sufficient condition for allroots of the characteristic polynomial to lie in the unit circle|z| < 1 on the complex plane is also derived. 2000 MathematicalsSubject Classification 39A11, 93D05.