Oscillation criteria for first-order neutral differential equations

This paper discusses a class of first-order neutral differential equations with variable coefficients and variable deviations. A series of sufficient conditions are established for all solutions of the equations to be oscillatory, and some of the conditions are sharp.     

一类变系数线性微分方程及其解

根据常系数线性微分方程的求解原理,通过一个适当变换,研究了一类变系数线性微分方程及其解的问题,从而可以得到这类方程在特征根都是互异单根时的解法和通解,并对三阶方程的各种情况进行了较为详尽的讨论.     

Dynamics and exact solutions of non-evolutionary partial differential equations

The article presents a new method for constructing exact solutions of non-evolutionary partial differential equations with two independent variables. The method is applied to the linear classical equations of mathematical physics: the Helmholtz equation and the variable type equation. The constructed method goes back to the theory of finite-dimensional dynamics proposed for evolutionary differential equations by B. Kruglikov, O. Lychagina and V. Lychagin. This theory is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of PDEs. The proposed method is used to construct exact particular solutions of linear differential equations (Helmholtz equations and equations of variable type).     

An indirect variable transformation approach and Jacobi elliptic solutions to Korteweg de Vries equation

Based on a variable change and the variable separated ODE method, an indirect variable transformation approach is proposed to search exact solutions to special types of partial differential equations (PDEs). The new method provides a more systematical and convenient handling of the solution process for the nonlinear equations. Its key point is to reduce the given PDEs to variable-coefficient ordinary differential equations, then we look for solutions to the resulting equations by some methods. As an application, exact solutions for the KdV equation are formally derived.     

Stability of solutions to differential equations with several variable delays. I

We consider a class of scalar linear differential equations with several variable delays and constant coefficients. A family of equations of the class is defined by coefficients and maximum admissible values of delays. We obtain conditions that are necessary and sufficient for the stability of solutions to all equations of the family. It is ascertained that the conditions are determined entirely by properties of the solution to the initial problem for an autonomous equation that belongs to the family. Some alternatives of required conditions are obtained in the form of estimates for solutions to autonomous equations in a finite interval.     

Some remarks on extremal solutions of multivalued differential equations

In this paper, we consider extremal solutions of multivalued differential equations, i.e., solutions that steer to the boundary of the attainable set. Multivalued differential equations arise in a natural way from control systems governed by ordinary differential equations that have a variable control-constraint set. Extremal solutions of multi-valued differential equations are important in the study of the optimal control of such systems. We give conditions under which extremality of a solution at a certain time implies extremality of the solution at all previous times where it is defined. Necessary conditions for extremality are also obtained. We treat both the time-dependent case and the time-independent case.     

Asymptotics of solutions of difference equations with delays

We consider a linear scalar difference equation with several variable delays and constant coefficients. The coefficients and maximum admissible values of delays are supposed to be the set of parameters that define a family of equations of the investigated class. We obtain effective necessary and sufficient conditions of the uniform and exponential stability of solutions to all equations of the family, as well as the conditions of the sign-definiteness and monotonicity of stable solutions.     

Burgers and Black–Merton–Scholes equations with real time variable and complex spatial variable

The purpose of this article is to study the Burgers and Black–Merton–Scholes equations with real time variable and complex spatial variable. The complexification of the spatial variable in these equations is made by two different methods which produce different equations: first, one complexifies the spatial variable in the corresponding (real) solution by replacing the usual sum of variables (translation) by an exponential product (rotation) and secondly, one complexifies the spatial variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions. By both methods, new kinds of evolution equations (or systems of equations) in two dimensional spatial variables are generated and their solutions are constructed.     

Die Polynomlösungen spezieller partieller Differentialgleichungen

One of the main problems in the theory of orthogonal polynomials in several variables is the determination of partial differential equations which have the given polynomials as their solutions. In this note, we consider partial differential equations which are two-dimensional generalizations of the classical differential equation for the Chebyshev polynomials in one variable and we will give conditions for its polynomial solutions. In addition, we will be able to determine all polynomials of a given class which are solutions of the partial differential equation under consideration. In the last section, we establish a connection between the different polynomial solutions.     

EXACT SOLUTIONS OF THE VARIABLE COEFFICIENT KdV AND SG TYPE EQUATIONS

In this paper,the variable cofficient KdV equation with dissipative loss and nonuniformity terms and the variable coefficient SG equation with nonuniformity term are studied. The exact solutions of the KdV and SG equations are obtained. In particular,the soliton solutions oftwo equations are found.     

一类高阶线性变系数常微分方程的通解

利用升阶法研究了一类高阶线性变系数常微分方程,给出了齐次方程的通解公式,并讨论了非齐次方程待定的特解.     

The exact solutions and the relevant constraint conditions for two nonlinear Schrödinger equations with variable coefficients

Two nonlinear Schrödinger equations with variable coefficients are researched, and the various exact solutions (including the bright and dark solitary waves) of the nonlinear Schrödinger equations are obtained with the aid of a subsidiary elliptic-like equation (sub-ODEs for short), at the same time, the constraint conditions which the coefficients of the nonlinear Schrödinger equations with variable coefficients satisfy are presented. The exact solutions and the constraint conditions are helpful in the application of the nonlinear Schrödinger equations with variable coefficients studied in this paper.     

Nonlinear matrix algebra and engineering applications. Part 1: Theory and linear form matrix

A matrix vector formalism is developed for systematizing the manipulation of sets of non-linear algebraic equations. In this formalism all manipulations are performed by multiplication with specially constructed transformation matrices. For many important classes of nonlinearities, algorithms based on this formalism are presented for rearranging a set of equations so that their solution may be obtained by numerically searching along a single variable. Theory developed proves that all solutions are obtained.     

Nonlinear matrix algebra and engineering applications Part 2: Polynomial form matrices

A matrix vector formalism is developed for systematizing the manipulation of sets of nonlinear algebraic equations. In this formalism all manipulations are performed by multiplication with specially constructed transformation matrices. For many important classes of nonlinearities, algorithms based on this formalism are presented for rearranging a set of equations so that their solution may be obtained by numerically searching along a single variable. Theory developed proves that all solutions are obtained.     

一类二阶常微分方程初值问题的无穷多解性

应用分离变量法,得到了一类二阶微分方程初值问题存在无穷多个非负解的充分必要条件,并给出了所有的无穷多个非负解．。     

Auxiliary equation method for solving nonlinear Wick-type partial differential equations

Using the solutions of an auxiliary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some Wick-type nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained. In addition, the links between Wick-type partial differential equations and variable coefficient partial differential equations are also clarified generally.     

三阶非线性中立型变时滞动力方程的渐近性与振动性(英文)

考虑了时标上三阶非线性中立型变时滞动力方程的渐近性和振动性.利用广义Ric-cati技巧与完全平方技巧,获得了方程所有解渐近和振动的准则.所得结果推广了已有三阶动力方程的结果,给出了一些例子加以说明本文的主要结论.     

ASYMPTOTICS AND OSCILLATION FOR FIRST ORDER MIXED NEUTRAL DIFFERENTIAL EQUATIONS

1IntroductionConsidertheneutraldifferentialequationswherepiERfori=l,2,...,m;o相似文献   

抽象空间中非线性算子方程变号解的存在性研究

利用Banach空间中的锥理论和不动点定理讨论了非线性算子方程变号解的存在性,给出了E_u_0空间下非线性算子方程变号解至少有一个变号解、一个正解和一个负解的条件,并讨论了仅通过一个上解条件得出非线性算子方程变号解的存在性定理.     

二阶变系数线性微分方程的几个可积类型

<正> 文[1]给出了一些变系数线性微分方程的可积类型,[2、3]利用“连锁”法也给出了一些变系数线性微分方程的可积类型,且包含了[1]的结果.本文继续讨论一般的二阶变系数线性微分方程