Asymptotics of Solutions of One Class of Linear Differential Equations of Arbitrary Order

In this paper, we study the problem on the asymptotic behavior of solutions of the equation l _n y = λy of arbitrary order (even or odd) with complex-valued coefficients in a neighborhood of the point +∞. The result obtained allows one to find defect numbers of the corresponding closed minimal symmetric differential operator.     

二阶周期线性微分方程解的几个性质

In this paper,the zeros of solutions of periodic second order linear differential equation y ＋ Ay = 0,where A（z） = B（e z ）,B（ζ） = g（ζ） ＋ p j=1 b ？j ζ ？j ,g（ζ） is a transcendental entire function of lower order no more than 1/2,and p is an odd positive integer,are studied.It is shown that every non-trivial solution of above equation satisfies the exponent of convergence of zeros equals to infinity.     

Properties A and B of nth Order Linear Differential Equations with Deviating Argument

Sufficient conditions for the nth order linear differential equation

高阶线性泛函方程的振动性

1 引言考虑高阶线性泛函方程其中 是给定的函数,x是未知实值函数. 表示函数g的m次迭代,即     

Bounded Oscillation of Nonlinear Neutral Differential Equations of Arbitrary Order

The paper is concerned with oscillation properties of n-th order neutral differential equations of the form

On the Dirichlet Problem in a Characteristic Rectangle for Fourth Order Linear Singular Hyperbolic Equations

In the rectangle D = (0,

线性微分方程的降阶法

提出线性微分方程的降阶法.在工科院校高数教材中若采用降阶法可以减少教材的篇幅.缩短教学时间,减轻学习难度.同时,还可利用数学软件求解线性微分方程,使高数教材贴近于现代计算技术.     

Nonoscillation Characterizations of Second Order Linear Differential Equations

In this paper, we establish some nonoscillatory characterizations of solutions of the second order linear differential equation (pu′)′ + qu = 0. Using these characterizations, we give some comparison theorems improving the results of Hille , Wintner , Wong and Yu .     

Properties of Complex Oscillation of Solutions of a Class of Higher Order Linear Differential Equations

Let A(z) be an entire function with μ(A) <1/2 such that the equation f⁽(k))+A(z)f = 0, where k ≥ 2, has a solution f with λ(f) < μ(A), and suppose that A₁ = A+h,where h■0 is an entire function with ρ(h) < μ(A). Then g⁽(k))+ A₁(z)g = 0 does not have a solution g with λ(g) < ∞.     

On the Correctness of the Dirichlet Problem in a Characteristic Rectangle for Fourth Order Linear Hyperbolic Equations

It is proved that the Dirichlet problem is correct in the characteristic rectangle D _ab = [0, a] × [0, b] for the linear hyperbolic equation

On Oscillation Criteria of Fourth Order Linear Differential Equations

The paper deals with oscillation criteria of fourth order linear differential equations with quasi-derivatives.     

On Periodic Solutions of Second Order Linear Difference Equations

In this paper, we apply a new procedure initially developed in Refs. [H. El-Owaidy and H.Y. Mohamed. "On the periodic solutions for nth order difference equations". Journal of Applied Mathematics and Computation , (to appear); "The necessary and sufficient conditions of existence of periodic solutions of nonautonomous difference equations". Journal of Applied Mathematics and Computation , (to appear)] to simplify the use of Carvalho's method to the case of discrete difference equations, in order to find the periodic solutions of second order linear difference equations. We can also find the complex periodic solutions.     

二阶半线性微分方程的振动定理

我们建立了二阶半线性微分方程(1)的新的振动定理,推广了Leighton和Kamenev的某些结果.     

Oscillation and Nonoscillation for Second Order Linear Differential Equations

New oscillation and nonoscillation theorems are obtained for the second order linear differential equationu″ + p(t)u = 0, wherep(t) ∈ C[0, ∞) andp(t) ≥ 0. Conditions only about the integrals ofp(t) on every interval [2ⁿt₀, 2^{n + 1}t₀] (n = 1, 2,…) for some fixedt₀ > 0 are used in the results.     

Linear Second Order Ordinary Differential Equations with Nonsmooth Coefficients

This paper deals with a Dirichlet boundary value problem for a linear second order ordinary differential operator, whose coefficients belong to certainL^p-spaces. Its solution is to be understood in the sense of Sobolev, so that the Fredholm alternative holds. The main purpose of this paper is, in case of unique solvability, to introduce a Green's function by means of which the solution can be given explicitly by integrals. We give the precise definition of the Green's function via Riesz' Representation Theorem and establish some of its basic properties. As a preliminary tool the Cauchy initial value problem is considered.     

High Order Optimized Geometric Integrators for Linear Differential Equations

In this paper new integration algorithms based on the Magnus expansion for linear differential equations up to eighth order are obtained. These methods are optimal with respect to the number of commutators required. Starting from Magnus series, integration schemes based on the Cayley transform an the Fer factorization are also built in terms of univariate integrals. The structure of the exact solution is retained while the computational cost is reduced compared to similar methods. Their relative performance is tested on some illustrative examples.     

Fourth Order Conservative Exponential Methods for Linear Evolution Equations

A few numerical methods for linear evolution equations are developed and analyzed in this paper. These fourth order exponential methods reproduce the exact solutions for equations with time-independent evolution operators. For highly oscillatory problems with evolution operators that vary slowly in time, these methods are often more efficient than the traditional methods, since large step sizes can be used. The methods developed in this paper are also conservative for equations such as the Schrödinger equation, where the evolution operator is skew-selfadjoint.     

Painlevé Classification of Binomial Type Ordinary Differential Equations of the Arbitrary Order

The complete Painlevé classification of the binomial ordinary differential equations of the arbitrary order n ≥ 4 is built. Six classes of equations with Painlevé property are obtained. All of these equations are solved in terms of elementary functions and known Painlevé transcendents.     

Integrable Systems of Linear Ordinary Differential Equations of General Order

In this paper, we define sufficient conditions for the solvability in quadratures of systems of linear ordinary differential equations of higher orders with variable coefficients. The method of variation of constants for equations of higher orders and for systems of equations of first order is generalized to the case of such systems. Results are obtained by reducing the systems under consideration to matrix equations of higher orders studied by methods of the theory of Lie algebras.