Asymptotic behavior and stability of second order neutral delay differential equations

We study the asymptotic behavior of a class of second order neutral delay differential equations by both a spectral projection method and an ordinary differential equation method approach. We discuss the relation of these two methods and illustrate some features using examples. Furthermore, a fixed point method is introduced as a third approach to study the asymptotic behavior. We conclude the paper with an application to a mechanical model of turning processes.     

System of differential equations with a strong turning point

The uniform asymptotics of a solution of a system of singularly perturbed differential equations with strong turning point is constructed. We study the case where the boundary operator is analytic with respect to a small parameter.     

On the exact WKB analysis of operators admitting infinitely many phases

To analyze differential operators whose WKB solutions admit infinitely many phases, we introduce a class of differential operators of WKB type and analyze their exact WKB theoretic structure near their turning points. Our analysis makes full use of techniques and ideas in microlocal analysis; we use a quantized contact transformation to construct a WKB solution of a differential equation of WKB type, and we use a Späth-type division theorem for a differential operator of WKB type to study its structure near turning points. As an application, we show a connection formula for WKB solutions near a simple turning point.     

关于具有转向点的一类常微分方程的边值问题

本文应用多重尺度法研究具有转向点的一类常微分方程的边值问题.避免了[1]中出现的悖理,以及[2]中关于确定任意常数的变分运算;构造出边值问题的解的一致有效渐近近似式.并研究了非共振的情形.     

Inner turning point in the theory of singular Perturbations

We construct the uniform asymptotics of a solution of a singularly perturbed differential equation of Liouville type with an interior turning point.     

Turning point in a system of differential equations with analytic operator

We construct uniform asymptotics for a solution of a system of singularly perturbed differential equations with turning point. We consider the case where the boundary operator analytically depends on a small parameter.     

Asymptotics of solutions of à system of differential equations with “turning points”

We construct an algorithm for determination of the principal term of the asymptotics of a solution of a system of differential equations with slow time and “turning points.”     

具有转向点的二阶非线性方程Robin问题的奇摄动(英)

本文应用微分不等式的方法，研究了具有转向点的二阶非线性方程Robin问题的奇摄动，根据fy在转向点附近的性态，边值问题的解将呈现冲击层现象和边界层现象，在适当的假设条件下，我们证明了该问题的存在性和不同的渐近性质。     

On the determination of differential equations with singularities and turning points

The inverse problem of synthesizing parameters of differential systems having a finite number of arbitrary order singularities and turning points is investigated. We establish properties of the spectral characteristics, prove a uniqueness theorem and provide a prodcedure for constructing the solution of the inverse problem.     

Inverse spectral problems for differential pencils on the half-line with turning points

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning points is studied. We establish properties of the spectral characteristics, give a formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.     

多约束二阶非线性常微分方程拐点的数值求法

提出了一种求解多约束二阶非线性常微分方程拐点的数值解法,并以具有平行反应的化学放热系统为例,给出了一个具体算例.     

解含转向点问题的完全指数型拟合差分方法

本文对含转向点的微分方程边值问题建立了完全指数型拟合差分格式,证明了此格式具有一阶一致收敛性.推广了Miller[1]的方法,简化了证明过程.数值结果表明本格式比Il'in[2]格式要好.     

Overstability: towards a global study

We consider singularly perturbed analytic differential equations in the complex domain with a turning point of order p and a p-dimensional control parameter a. We prove existence and exponential closeness of well behaved solutions under some geometric and transversality conditions. Furthermore, properties of formal solutions are studied and an application to “least term summation” is given.     

Pathfollowing for essentially singular boundary value problems with application to the complex Ginzburg-Landau equation

We present a pathfollowing strategy based on pseudo-arclength parametrization for the solution of parameter-dependent boundary value problems for ordinary differential equations. We formulate criteria which ensure the successful application of this method for the computation of solution branches with turning points for problems with an essential singularity. The advantages of our approach result from the possibility to use efficient mesh selection, and a favorable conditioning even for problems posed on a semi-infinite interval and subsequently transformed to an essentially singular problem. This is demonstrated by a Matlab implementation of the solution method based on an adaptive collocation scheme which is well suited to solve problems of practical relevance. As one example, we compute solution branches for the complex Ginzburg-Landau equation which start from non-monotone ‘multi-bump’ solutions of the nonlinear Schrödinger equation. Following the branches around turning points, real-valued solutions of the nonlinear Schrödinger equation can easily be computed.     

具有转向点的二阶非线性系统Robin边值问题的奇摄动

本文研究一类具有转向点的二阶非线性系统 Robin边值问题的奇摄动 ,在适当的假设条件下 ,利用微分不等式方法证明了解的存在性 ,并得到了解的按分量的渐近估计式 .     

具有转向点的三阶半线性奇摄动边值问题解的存在性

本文应用微分不等式技术证明一类具有转向点的三阶半线性奇摄动边值问题解的存在性.     

SINGULAR PERTURBATION OF THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEM WITH TWO TURNING POINTS

1IntroductionManyauthorshavedealtwiththeboundaryvalueproblemsofsecondorthirdorderdifferentialequationwithaturningpoint[l-5],butthatwithtwoormoreturningpointsisscarcelydiscussed.ThispaperconcernsthefollowingboundaryvalueproblemwhereE>0isasmallparameter,0,I…     

Linear-quadratic control and quadratic differential forms for multidimensional behaviors

This paper deals with systems described by constant coefficient linear partial differential equations (nD-systems) from a behavioral point of view. In this context we treat the linear quadratic control problem where the performance functional is the integral of a quadratic differential form. We look for characterizations of the set of stationary trajectories and of the set of local minimal trajectories with respect to compact support variations, turning out that they are equal if the system is dissipative. Finally we provide conditions for regular implementability of this set of trajectories and give an explicit representation of an optimal controller.     

The Adomian decomposition method in turning point problems

In this paper, the decomposition method is applied to boundary-value problems of ordinary differential equations with a parameter exhibiting turning points.     

Libera transform of functions with bounded turning

Conditions are determined for the starlikeness of the Libera transform of functions of bounded turning. In addition, several other differential subordinations and differential inequalities are considered.