一类二阶变系数非齐次线性微分方程的通解

通过在二阶变系数非齐次线性微分方程两边同乘以某个积分因子将该方程转化为常系数非齐次线性微分方程,进而得出二阶变系数非齐次线性微分方程的通解公式.     

一些特殊类型的变系数二阶线性微分方程解法的研究

运用变量变换的方法将一些特殊类型的变系数二阶线性微分方程化为常系数二阶线性微分方程,或已知齐次方程的一个解来求出齐次方程的另一个线性无关解,从而达到按照常系数二阶线性微分方程的特殊方法和利用常数变易法来求方程的通解的目的,同时纠正了文献[3]的结论和例子2的错误.     

变系数二阶线性微分方程一个新的可解类型再讨论

利用二阶线性微分方程的不变量,给出二阶线性微分方程常系数与变系数、齐次与非齐次的统一解法,而且扩大了自由项函数的形式.     

二阶变系数线性微分方程的解法探讨

给出了变系数满足几种特定条件的二阶变系数齐次线性微分方程的特解形式,得到了一个命题.之后通过几个典型实例验证了命题在求解几类二阶变系数线性微分方程特解和通解中的有效性.     

特殊类型二阶变系数非齐次线性微分方程通解公式

本文给出了二阶变系数非齐次线性微分方程的系数满足某些特殊条件时的通解公式     

浅析二阶齐次线性变系数微分方程的一个可积类型

本文讨论了二阶齐次线性变系数微分方程的特殊形式，给出了这种微分方程的一个可积类型．     

亚纯系数的高阶线性微分方程的解的增长性

本文研究亚纯系数的高阶线性微分方程,当方程系数满足一定条件时,得到方程的每一非零亚纯解具有无穷级且超级为n.此外,还研究了非齐次线性微分方程的亚纯解.     

特殊类型二阶变系数非齐次线性微分方程通解公式

本给出了二阶变系数非齐次线性微分方程的系数满足某些特殊条件时的通解公式．     

一类二阶线性变系数微分方程通解的解法

采用特解和常数变易法,给出一类二阶线性变系数齐次和非齐次微分方程的通解公式,实例说明如何运用此通解公式.     

单位圆内二阶线性微分方程的解与小函数的关系

讨论了系数是单位圆内解析函数的二阶齐次和非齐次线性微分方程的解及其一阶导数和二阶导数与小函数之间的关系,并得到了它们之间的精确估计.     

A generalization of the Schwarzian derivative and its applications

A generalization of the Schwarzian derivative is proposed; this generalization is used to investigate functions having divided differences which are nonzero in a given region. A relation is established between this class of functions and linear homogeneous differential equations.Translated from Matematicheskie Zametki, Vol. 10, No. 2, pp. 229–238, August, 1971.     

The method of approximate particular solutions for solving certain partial differential equations

A standard approach for solving linear partial differential equations is to split the solution into a homogeneous solution and a particular solution. Motivated by the method of fundamental solutions for solving homogeneous equations, we propose a similar approach using the method of approximate particular solutions for solving linear inhomogeneous differential equations without the need of finding the homogeneous solution. This leads to a much simpler numerical scheme with similar accuracy to the traditional approach. To demonstrate the simplicity of the new approach, three numerical examples are given with excellent results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 506–522, 2012     

Existence of Small Periodic Solutions of Nonlinear Systems of Ordinary Differential Equations

We investigate the case where conditions for the existence of a nonzero periodic solution of a system of ordinary differential equations are determined by the properties of elements of the matrix of linear approximation and the properties of nonlinear terms.     

An effective boundary element method for inhomogeneous partial differential equations

A method for removing the domain or volume integral arising in boundary integral formulations for linear inhomogeneous partial differential equations is presented. The technique removes the integral by considering a particular solution to the homogeneous partial differential equation which approximates the inhomogeneity in terms of radial basis functions. The remainder of the solution will then satisfy a homogeneous partial differential equation and hence lead to an integral equation with only boundary contributions. Some results for the inhomogeneous Poisson equation and for linear elastostatics with known body forces are presented.     

常系数非齐次线性微分方程组的初等解法

利用初等变换将常系数非齐次线性微分方程组化为由若干个相互独立的高阶常系数非齐次线性微分方程组成的方程组,再利用高阶常系数齐次线性微分方程的特征根法和非齐次方程的待定系数法求该方程组的基本解组及特解,最后通过初等变换求原方程组的基本解组及特解,从而可求出其通解.     

On the optimal recovery of a solution of a system of linear homogeneous differential equations

We consider the optimal recovery problem for a solution of a system of linear homogeneous ordinary differential equations with constant self-adjoint coefficient matrix on the basis of inexactly given values of the solution at two time instants.     

On nonzero initial conditions in stochastic differential equations

The effect of nonzero initial conditions in Adomian's solution of linear, or nonlinear, stochastic differential equations is demonstrated for an Nth-order equation in which the decomposition of the linear part into L+R is made so as to simplify the Green's function.     

Lyapunov exponent,Liao perturbation and persistence Dedicated to the Memory of Professor Shantao Liao at the Centenary of His Birth

Consider a C~1 vector field together with an ergodic invariant probability that has ? nonzero Lyapunov exponents. Using orthonormal moving frames along a generic orbit we construct a linear system of ?differential equations which is a linearized Liao standard system. We show that Lyapunov exponents of this linear system coincide with all the nonzero exponents of the given vector field with respect to the given ergodic probability. Moreover, we prove that these Lyapunov exponents have a persistence property meaning that a small perturbation to the linear system(Liao perturbation) preserves both the sign and the value of the nonzero Lyapunov exponents.     

Fourier–Bessel theory on flow acoustics in inviscid shear pipeline fluid flow

Flow acoustics in pipeline is of considerable interest in both industrial application and scientific research. While well-known analytical solutions exist for stationary and uniform mean flow, only numerical solutions exist for shear mean flow. Based on potential theory, a general mathematical formulation of flow acoustics in inviscid fluid with shear mean flow is deduced, resulting in a set of two second-order differential equations. According to Fourier–Bessel theory which is orthogonal and complete in Lebesgue Space, a solution is proposed to transform the differential equations to linear homogeneous algebraic equations. Consequently, the axial wave number is numerically calculated due to the existence condition of non-trivial solution to homogeneous linear algebraic equations, leading to the vanishment of the corresponding determinant. Based on the proposed method, wave propagation in laminar and turbulent flow is numerically analyzed.     

On the resolvent of linear nonautonomous partial functional differential equations

We give sufficient conditions for the existence of the resolvent operator for nonautonomous linear partial differential equations with delay, where the highest order derivatives are undelayed. Furthermore we analyse the connection between the resolvent and the solution operator of the homogeneous equation.